2000 character limit reached
Schur-Weyl dualities for the rook monoid: an approach via Schur algebras (2404.01493v1)
Published 1 Apr 2024 in math.RT
Abstract: The rook monoid, also known as the symmetric inverse monoid, is the archetypal structure when it comes to extend the principle of symmetry. In this paper, we establish a Schur-Weyl duality between this monoid and an extension of the classical Schur algebra, which we name the extended Schur algebra. We also explain how this relates to Solomon's Schur-Weyl duality between the rook monoid and the general linear group and mention some advantages of our approach.
- Bowman C, Doty S, Martin S (2022) Integral Schur-Weyl duality for partition algebras. Algebr Comb 5(2):371–399. 10.5802/alco.214, URL https://doi.org/10.5802/alco.214 Brauer [1937] Brauer R (1937) On algebras which are connected with the semisimple continuous groups. Ann of Math (2) 38(4):857–872. 10.2307/1968843, URL http://dx.doi.org/10.2307/1968843 Carter and Lusztig [1974] Carter RW, Lusztig G (1974) On the modular representations of the general linear and symmetric groups. Math Z 136:193–242 de Concini and Procesi [1976] de Concini C, Procesi C (1976) A characteristic free approach to invariant theory. Advances in Math 21(3):330–354 Curtis and Reiner [1981] Curtis CW, Reiner I (1981) Methods of representation theory. Vol. I. John Wiley & Sons Inc., New York, with applications to finite groups and orders, Pure and Applied Mathematics, A Wiley-Interscience Publication Dipper et al [2008] Dipper R, Doty S, Hu J (2008) Brauer algebras, symplectic Schur algebras and Schur-Weyl duality. Trans Amer Math Soc 360(1):189–213. 10.1090/S0002-9947-07-04179-7, URL https://doi.org/10.1090/S0002-9947-07-04179-7 Doty and Hu [2009] Doty S, Hu J (2009) Schur-Weyl duality for orthogonal groups. Proc Lond Math Soc (3) 98(3):679–713. 10.1112/plms/pdn044, URL https://doi.org/10.1112/plms/pdn044 Goodman and Wallach [2009] Goodman R, Wallach NR (2009) Symmetry, representations, and invariants, Graduate Texts in Mathematics, vol 255. Springer, Dordrecht, 10.1007/978-0-387-79852-3, URL https://doi.org/10.1007/978-0-387-79852-3 Graham and Lehrer [1996] Graham JJ, Lehrer GI (1996) Cellular algebras. Invent Math 123(1):1–34. 10.1007/BF01232365, URL http://dx.doi.org/10.1007/BF01232365 Green [1980] Green JA (1980) Polynomial representations of GLnsubscriptGL𝑛{\rm GL}_{n}roman_GL start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT, Lecture Notes in Mathematics, vol 830. Springer-Verlag, Berlin Grood [2006] Grood C (2006) The rook partition algebra. J Combin Theory Ser A 113(2):325–351. 10.1016/j.jcta.2005.03.006, URL https://doi.org/10.1016/j.jcta.2005.03.006 Halverson [1996] Halverson T (1996) Characters of the centralizer algebras of mixed tensor representations of Gl(r,ℂ)Gl𝑟ℂ{\rm Gl}(r,\mathbb{C})roman_Gl ( italic_r , blackboard_C ) and the quantum group 𝔘q(ℊl(r,ℂ))subscript𝔘𝑞ℊ𝑙𝑟ℂ\mathfrak{U}_{q}({\mathcal{g}l}(r,\mathbb{C}))fraktur_U start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ( caligraphic_g italic_l ( italic_r , blackboard_C ) ). Pacific J Math 174(2):359–410. URL http://projecteuclid.org/euclid.pjm/1102365176 Halverson [2004] Halverson T (2004) Representations of the q𝑞qitalic_q-rook monoid. J Algebra 273(1):227–251. 10.1016/j.jalgebra.2003.11.002, URL http://dx.doi.org/10.1016/j.jalgebra.2003.11.002 Halverson and delMas [2014] Halverson T, delMas E (2014) Representations of the Rook-Brauer algebra. Comm Algebra 42(1):423–443. 10.1080/00927872.2012.716120, URL https://doi.org/10.1080/00927872.2012.716120 Halverson and Ram [2005] Halverson T, Ram A (2005) Partition algebras. European J Combin 26(6):869–921. 10.1016/j.ejc.2004.06.005, URL http://dx.doi.org/10.1016/j.ejc.2004.06.005 Jacobson [1989] Jacobson N (1989) Basic algebra. II, 2nd edn. W. H. Freeman and Company, New York Jones [1994] Jones VFR (1994) The Potts model and the symmetric group. In: Subfactors (Kyuzeso, 1993). World Sci. Publ., River Edge, NJ, p 259–267 Kudryavtseva and Mazorchuk [2008] Kudryavtseva G, Mazorchuk V (2008) Schur-Weyl dualities for symmetric inverse semigroups. J Pure Appl Algebra 212(8):1987–1995. 10.1016/j.jpaa.2007.12.004, URL https://doi.org/10.1016/j.jpaa.2007.12.004 Lawson [1998] Lawson MV (1998) Inverse semigroups. World Scientific Publishing Co. Inc., River Edge, NJ, 10.1142/9789812816689, URL http://dx.doi.org/10.1142/9789812816689, the theory of partial symmetries Martin [1994] Martin P (1994) Temperley-Lieb algebras for nonplanar statistical mechanics—the partition algebra construction. J Knot Theory Ramifications 3(1):51–82. 10.1142/S0218216594000071, URL http://dx.doi.org/10.1142/S0218216594000071 Martin [1996] Martin P (1996) The structure of the partition algebras. J Algebra 183(2):319–358. 10.1006/jabr.1996.0223, URL http://dx.doi.org/10.1006/jabr.1996.0223 Martin and Mazorchuk [2014] Martin P, Mazorchuk V (2014) On the representation theory of partial Brauer algebras. Q J Math 65(1):225–247. 10.1093/qmath/has043, URL https://doi.org/10.1093/qmath/has043 Martin [1990] Martin PP (1990) Representations of graph Temperley-Lieb algebras. Publ Res Inst Math Sci 26(3):485–503. 10.2977/prims/1195170958, URL https://doi.org/10.2977/prims/1195170958 Martin [2008] Martin S (2008) Schur algebras and representation theory, Cambridge Tracts in Mathematics, vol 112. Cambridge University Press, Cambridge, reprint of the 1993 original Munn [1955] Munn WD (1955) On semigroup algebras. Proc Cambridge Philos Soc 51:1–15 Munn [1957a] Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Brauer R (1937) On algebras which are connected with the semisimple continuous groups. Ann of Math (2) 38(4):857–872. 10.2307/1968843, URL http://dx.doi.org/10.2307/1968843 Carter and Lusztig [1974] Carter RW, Lusztig G (1974) On the modular representations of the general linear and symmetric groups. Math Z 136:193–242 de Concini and Procesi [1976] de Concini C, Procesi C (1976) A characteristic free approach to invariant theory. Advances in Math 21(3):330–354 Curtis and Reiner [1981] Curtis CW, Reiner I (1981) Methods of representation theory. Vol. I. John Wiley & Sons Inc., New York, with applications to finite groups and orders, Pure and Applied Mathematics, A Wiley-Interscience Publication Dipper et al [2008] Dipper R, Doty S, Hu J (2008) Brauer algebras, symplectic Schur algebras and Schur-Weyl duality. Trans Amer Math Soc 360(1):189–213. 10.1090/S0002-9947-07-04179-7, URL https://doi.org/10.1090/S0002-9947-07-04179-7 Doty and Hu [2009] Doty S, Hu J (2009) Schur-Weyl duality for orthogonal groups. Proc Lond Math Soc (3) 98(3):679–713. 10.1112/plms/pdn044, URL https://doi.org/10.1112/plms/pdn044 Goodman and Wallach [2009] Goodman R, Wallach NR (2009) Symmetry, representations, and invariants, Graduate Texts in Mathematics, vol 255. Springer, Dordrecht, 10.1007/978-0-387-79852-3, URL https://doi.org/10.1007/978-0-387-79852-3 Graham and Lehrer [1996] Graham JJ, Lehrer GI (1996) Cellular algebras. Invent Math 123(1):1–34. 10.1007/BF01232365, URL http://dx.doi.org/10.1007/BF01232365 Green [1980] Green JA (1980) Polynomial representations of GLnsubscriptGL𝑛{\rm GL}_{n}roman_GL start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT, Lecture Notes in Mathematics, vol 830. Springer-Verlag, Berlin Grood [2006] Grood C (2006) The rook partition algebra. J Combin Theory Ser A 113(2):325–351. 10.1016/j.jcta.2005.03.006, URL https://doi.org/10.1016/j.jcta.2005.03.006 Halverson [1996] Halverson T (1996) Characters of the centralizer algebras of mixed tensor representations of Gl(r,ℂ)Gl𝑟ℂ{\rm Gl}(r,\mathbb{C})roman_Gl ( italic_r , blackboard_C ) and the quantum group 𝔘q(ℊl(r,ℂ))subscript𝔘𝑞ℊ𝑙𝑟ℂ\mathfrak{U}_{q}({\mathcal{g}l}(r,\mathbb{C}))fraktur_U start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ( caligraphic_g italic_l ( italic_r , blackboard_C ) ). Pacific J Math 174(2):359–410. URL http://projecteuclid.org/euclid.pjm/1102365176 Halverson [2004] Halverson T (2004) Representations of the q𝑞qitalic_q-rook monoid. J Algebra 273(1):227–251. 10.1016/j.jalgebra.2003.11.002, URL http://dx.doi.org/10.1016/j.jalgebra.2003.11.002 Halverson and delMas [2014] Halverson T, delMas E (2014) Representations of the Rook-Brauer algebra. Comm Algebra 42(1):423–443. 10.1080/00927872.2012.716120, URL https://doi.org/10.1080/00927872.2012.716120 Halverson and Ram [2005] Halverson T, Ram A (2005) Partition algebras. European J Combin 26(6):869–921. 10.1016/j.ejc.2004.06.005, URL http://dx.doi.org/10.1016/j.ejc.2004.06.005 Jacobson [1989] Jacobson N (1989) Basic algebra. II, 2nd edn. W. H. Freeman and Company, New York Jones [1994] Jones VFR (1994) The Potts model and the symmetric group. In: Subfactors (Kyuzeso, 1993). World Sci. Publ., River Edge, NJ, p 259–267 Kudryavtseva and Mazorchuk [2008] Kudryavtseva G, Mazorchuk V (2008) Schur-Weyl dualities for symmetric inverse semigroups. J Pure Appl Algebra 212(8):1987–1995. 10.1016/j.jpaa.2007.12.004, URL https://doi.org/10.1016/j.jpaa.2007.12.004 Lawson [1998] Lawson MV (1998) Inverse semigroups. World Scientific Publishing Co. Inc., River Edge, NJ, 10.1142/9789812816689, URL http://dx.doi.org/10.1142/9789812816689, the theory of partial symmetries Martin [1994] Martin P (1994) Temperley-Lieb algebras for nonplanar statistical mechanics—the partition algebra construction. J Knot Theory Ramifications 3(1):51–82. 10.1142/S0218216594000071, URL http://dx.doi.org/10.1142/S0218216594000071 Martin [1996] Martin P (1996) The structure of the partition algebras. J Algebra 183(2):319–358. 10.1006/jabr.1996.0223, URL http://dx.doi.org/10.1006/jabr.1996.0223 Martin and Mazorchuk [2014] Martin P, Mazorchuk V (2014) On the representation theory of partial Brauer algebras. Q J Math 65(1):225–247. 10.1093/qmath/has043, URL https://doi.org/10.1093/qmath/has043 Martin [1990] Martin PP (1990) Representations of graph Temperley-Lieb algebras. Publ Res Inst Math Sci 26(3):485–503. 10.2977/prims/1195170958, URL https://doi.org/10.2977/prims/1195170958 Martin [2008] Martin S (2008) Schur algebras and representation theory, Cambridge Tracts in Mathematics, vol 112. Cambridge University Press, Cambridge, reprint of the 1993 original Munn [1955] Munn WD (1955) On semigroup algebras. Proc Cambridge Philos Soc 51:1–15 Munn [1957a] Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Carter RW, Lusztig G (1974) On the modular representations of the general linear and symmetric groups. Math Z 136:193–242 de Concini and Procesi [1976] de Concini C, Procesi C (1976) A characteristic free approach to invariant theory. Advances in Math 21(3):330–354 Curtis and Reiner [1981] Curtis CW, Reiner I (1981) Methods of representation theory. Vol. I. John Wiley & Sons Inc., New York, with applications to finite groups and orders, Pure and Applied Mathematics, A Wiley-Interscience Publication Dipper et al [2008] Dipper R, Doty S, Hu J (2008) Brauer algebras, symplectic Schur algebras and Schur-Weyl duality. Trans Amer Math Soc 360(1):189–213. 10.1090/S0002-9947-07-04179-7, URL https://doi.org/10.1090/S0002-9947-07-04179-7 Doty and Hu [2009] Doty S, Hu J (2009) Schur-Weyl duality for orthogonal groups. Proc Lond Math Soc (3) 98(3):679–713. 10.1112/plms/pdn044, URL https://doi.org/10.1112/plms/pdn044 Goodman and Wallach [2009] Goodman R, Wallach NR (2009) Symmetry, representations, and invariants, Graduate Texts in Mathematics, vol 255. Springer, Dordrecht, 10.1007/978-0-387-79852-3, URL https://doi.org/10.1007/978-0-387-79852-3 Graham and Lehrer [1996] Graham JJ, Lehrer GI (1996) Cellular algebras. Invent Math 123(1):1–34. 10.1007/BF01232365, URL http://dx.doi.org/10.1007/BF01232365 Green [1980] Green JA (1980) Polynomial representations of GLnsubscriptGL𝑛{\rm GL}_{n}roman_GL start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT, Lecture Notes in Mathematics, vol 830. Springer-Verlag, Berlin Grood [2006] Grood C (2006) The rook partition algebra. J Combin Theory Ser A 113(2):325–351. 10.1016/j.jcta.2005.03.006, URL https://doi.org/10.1016/j.jcta.2005.03.006 Halverson [1996] Halverson T (1996) Characters of the centralizer algebras of mixed tensor representations of Gl(r,ℂ)Gl𝑟ℂ{\rm Gl}(r,\mathbb{C})roman_Gl ( italic_r , blackboard_C ) and the quantum group 𝔘q(ℊl(r,ℂ))subscript𝔘𝑞ℊ𝑙𝑟ℂ\mathfrak{U}_{q}({\mathcal{g}l}(r,\mathbb{C}))fraktur_U start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ( caligraphic_g italic_l ( italic_r , blackboard_C ) ). Pacific J Math 174(2):359–410. URL http://projecteuclid.org/euclid.pjm/1102365176 Halverson [2004] Halverson T (2004) Representations of the q𝑞qitalic_q-rook monoid. J Algebra 273(1):227–251. 10.1016/j.jalgebra.2003.11.002, URL http://dx.doi.org/10.1016/j.jalgebra.2003.11.002 Halverson and delMas [2014] Halverson T, delMas E (2014) Representations of the Rook-Brauer algebra. Comm Algebra 42(1):423–443. 10.1080/00927872.2012.716120, URL https://doi.org/10.1080/00927872.2012.716120 Halverson and Ram [2005] Halverson T, Ram A (2005) Partition algebras. European J Combin 26(6):869–921. 10.1016/j.ejc.2004.06.005, URL http://dx.doi.org/10.1016/j.ejc.2004.06.005 Jacobson [1989] Jacobson N (1989) Basic algebra. II, 2nd edn. W. H. Freeman and Company, New York Jones [1994] Jones VFR (1994) The Potts model and the symmetric group. In: Subfactors (Kyuzeso, 1993). World Sci. Publ., River Edge, NJ, p 259–267 Kudryavtseva and Mazorchuk [2008] Kudryavtseva G, Mazorchuk V (2008) Schur-Weyl dualities for symmetric inverse semigroups. J Pure Appl Algebra 212(8):1987–1995. 10.1016/j.jpaa.2007.12.004, URL https://doi.org/10.1016/j.jpaa.2007.12.004 Lawson [1998] Lawson MV (1998) Inverse semigroups. World Scientific Publishing Co. Inc., River Edge, NJ, 10.1142/9789812816689, URL http://dx.doi.org/10.1142/9789812816689, the theory of partial symmetries Martin [1994] Martin P (1994) Temperley-Lieb algebras for nonplanar statistical mechanics—the partition algebra construction. J Knot Theory Ramifications 3(1):51–82. 10.1142/S0218216594000071, URL http://dx.doi.org/10.1142/S0218216594000071 Martin [1996] Martin P (1996) The structure of the partition algebras. J Algebra 183(2):319–358. 10.1006/jabr.1996.0223, URL http://dx.doi.org/10.1006/jabr.1996.0223 Martin and Mazorchuk [2014] Martin P, Mazorchuk V (2014) On the representation theory of partial Brauer algebras. Q J Math 65(1):225–247. 10.1093/qmath/has043, URL https://doi.org/10.1093/qmath/has043 Martin [1990] Martin PP (1990) Representations of graph Temperley-Lieb algebras. Publ Res Inst Math Sci 26(3):485–503. 10.2977/prims/1195170958, URL https://doi.org/10.2977/prims/1195170958 Martin [2008] Martin S (2008) Schur algebras and representation theory, Cambridge Tracts in Mathematics, vol 112. Cambridge University Press, Cambridge, reprint of the 1993 original Munn [1955] Munn WD (1955) On semigroup algebras. Proc Cambridge Philos Soc 51:1–15 Munn [1957a] Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. de Concini C, Procesi C (1976) A characteristic free approach to invariant theory. Advances in Math 21(3):330–354 Curtis and Reiner [1981] Curtis CW, Reiner I (1981) Methods of representation theory. Vol. I. John Wiley & Sons Inc., New York, with applications to finite groups and orders, Pure and Applied Mathematics, A Wiley-Interscience Publication Dipper et al [2008] Dipper R, Doty S, Hu J (2008) Brauer algebras, symplectic Schur algebras and Schur-Weyl duality. Trans Amer Math Soc 360(1):189–213. 10.1090/S0002-9947-07-04179-7, URL https://doi.org/10.1090/S0002-9947-07-04179-7 Doty and Hu [2009] Doty S, Hu J (2009) Schur-Weyl duality for orthogonal groups. Proc Lond Math Soc (3) 98(3):679–713. 10.1112/plms/pdn044, URL https://doi.org/10.1112/plms/pdn044 Goodman and Wallach [2009] Goodman R, Wallach NR (2009) Symmetry, representations, and invariants, Graduate Texts in Mathematics, vol 255. Springer, Dordrecht, 10.1007/978-0-387-79852-3, URL https://doi.org/10.1007/978-0-387-79852-3 Graham and Lehrer [1996] Graham JJ, Lehrer GI (1996) Cellular algebras. Invent Math 123(1):1–34. 10.1007/BF01232365, URL http://dx.doi.org/10.1007/BF01232365 Green [1980] Green JA (1980) Polynomial representations of GLnsubscriptGL𝑛{\rm GL}_{n}roman_GL start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT, Lecture Notes in Mathematics, vol 830. Springer-Verlag, Berlin Grood [2006] Grood C (2006) The rook partition algebra. J Combin Theory Ser A 113(2):325–351. 10.1016/j.jcta.2005.03.006, URL https://doi.org/10.1016/j.jcta.2005.03.006 Halverson [1996] Halverson T (1996) Characters of the centralizer algebras of mixed tensor representations of Gl(r,ℂ)Gl𝑟ℂ{\rm Gl}(r,\mathbb{C})roman_Gl ( italic_r , blackboard_C ) and the quantum group 𝔘q(ℊl(r,ℂ))subscript𝔘𝑞ℊ𝑙𝑟ℂ\mathfrak{U}_{q}({\mathcal{g}l}(r,\mathbb{C}))fraktur_U start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ( caligraphic_g italic_l ( italic_r , blackboard_C ) ). Pacific J Math 174(2):359–410. URL http://projecteuclid.org/euclid.pjm/1102365176 Halverson [2004] Halverson T (2004) Representations of the q𝑞qitalic_q-rook monoid. J Algebra 273(1):227–251. 10.1016/j.jalgebra.2003.11.002, URL http://dx.doi.org/10.1016/j.jalgebra.2003.11.002 Halverson and delMas [2014] Halverson T, delMas E (2014) Representations of the Rook-Brauer algebra. Comm Algebra 42(1):423–443. 10.1080/00927872.2012.716120, URL https://doi.org/10.1080/00927872.2012.716120 Halverson and Ram [2005] Halverson T, Ram A (2005) Partition algebras. European J Combin 26(6):869–921. 10.1016/j.ejc.2004.06.005, URL http://dx.doi.org/10.1016/j.ejc.2004.06.005 Jacobson [1989] Jacobson N (1989) Basic algebra. II, 2nd edn. W. H. Freeman and Company, New York Jones [1994] Jones VFR (1994) The Potts model and the symmetric group. In: Subfactors (Kyuzeso, 1993). World Sci. Publ., River Edge, NJ, p 259–267 Kudryavtseva and Mazorchuk [2008] Kudryavtseva G, Mazorchuk V (2008) Schur-Weyl dualities for symmetric inverse semigroups. J Pure Appl Algebra 212(8):1987–1995. 10.1016/j.jpaa.2007.12.004, URL https://doi.org/10.1016/j.jpaa.2007.12.004 Lawson [1998] Lawson MV (1998) Inverse semigroups. World Scientific Publishing Co. Inc., River Edge, NJ, 10.1142/9789812816689, URL http://dx.doi.org/10.1142/9789812816689, the theory of partial symmetries Martin [1994] Martin P (1994) Temperley-Lieb algebras for nonplanar statistical mechanics—the partition algebra construction. J Knot Theory Ramifications 3(1):51–82. 10.1142/S0218216594000071, URL http://dx.doi.org/10.1142/S0218216594000071 Martin [1996] Martin P (1996) The structure of the partition algebras. J Algebra 183(2):319–358. 10.1006/jabr.1996.0223, URL http://dx.doi.org/10.1006/jabr.1996.0223 Martin and Mazorchuk [2014] Martin P, Mazorchuk V (2014) On the representation theory of partial Brauer algebras. Q J Math 65(1):225–247. 10.1093/qmath/has043, URL https://doi.org/10.1093/qmath/has043 Martin [1990] Martin PP (1990) Representations of graph Temperley-Lieb algebras. Publ Res Inst Math Sci 26(3):485–503. 10.2977/prims/1195170958, URL https://doi.org/10.2977/prims/1195170958 Martin [2008] Martin S (2008) Schur algebras and representation theory, Cambridge Tracts in Mathematics, vol 112. Cambridge University Press, Cambridge, reprint of the 1993 original Munn [1955] Munn WD (1955) On semigroup algebras. Proc Cambridge Philos Soc 51:1–15 Munn [1957a] Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Curtis CW, Reiner I (1981) Methods of representation theory. Vol. I. John Wiley & Sons Inc., New York, with applications to finite groups and orders, Pure and Applied Mathematics, A Wiley-Interscience Publication Dipper et al [2008] Dipper R, Doty S, Hu J (2008) Brauer algebras, symplectic Schur algebras and Schur-Weyl duality. Trans Amer Math Soc 360(1):189–213. 10.1090/S0002-9947-07-04179-7, URL https://doi.org/10.1090/S0002-9947-07-04179-7 Doty and Hu [2009] Doty S, Hu J (2009) Schur-Weyl duality for orthogonal groups. Proc Lond Math Soc (3) 98(3):679–713. 10.1112/plms/pdn044, URL https://doi.org/10.1112/plms/pdn044 Goodman and Wallach [2009] Goodman R, Wallach NR (2009) Symmetry, representations, and invariants, Graduate Texts in Mathematics, vol 255. Springer, Dordrecht, 10.1007/978-0-387-79852-3, URL https://doi.org/10.1007/978-0-387-79852-3 Graham and Lehrer [1996] Graham JJ, Lehrer GI (1996) Cellular algebras. Invent Math 123(1):1–34. 10.1007/BF01232365, URL http://dx.doi.org/10.1007/BF01232365 Green [1980] Green JA (1980) Polynomial representations of GLnsubscriptGL𝑛{\rm GL}_{n}roman_GL start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT, Lecture Notes in Mathematics, vol 830. Springer-Verlag, Berlin Grood [2006] Grood C (2006) The rook partition algebra. J Combin Theory Ser A 113(2):325–351. 10.1016/j.jcta.2005.03.006, URL https://doi.org/10.1016/j.jcta.2005.03.006 Halverson [1996] Halverson T (1996) Characters of the centralizer algebras of mixed tensor representations of Gl(r,ℂ)Gl𝑟ℂ{\rm Gl}(r,\mathbb{C})roman_Gl ( italic_r , blackboard_C ) and the quantum group 𝔘q(ℊl(r,ℂ))subscript𝔘𝑞ℊ𝑙𝑟ℂ\mathfrak{U}_{q}({\mathcal{g}l}(r,\mathbb{C}))fraktur_U start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ( caligraphic_g italic_l ( italic_r , blackboard_C ) ). Pacific J Math 174(2):359–410. URL http://projecteuclid.org/euclid.pjm/1102365176 Halverson [2004] Halverson T (2004) Representations of the q𝑞qitalic_q-rook monoid. J Algebra 273(1):227–251. 10.1016/j.jalgebra.2003.11.002, URL http://dx.doi.org/10.1016/j.jalgebra.2003.11.002 Halverson and delMas [2014] Halverson T, delMas E (2014) Representations of the Rook-Brauer algebra. Comm Algebra 42(1):423–443. 10.1080/00927872.2012.716120, URL https://doi.org/10.1080/00927872.2012.716120 Halverson and Ram [2005] Halverson T, Ram A (2005) Partition algebras. European J Combin 26(6):869–921. 10.1016/j.ejc.2004.06.005, URL http://dx.doi.org/10.1016/j.ejc.2004.06.005 Jacobson [1989] Jacobson N (1989) Basic algebra. II, 2nd edn. W. H. Freeman and Company, New York Jones [1994] Jones VFR (1994) The Potts model and the symmetric group. In: Subfactors (Kyuzeso, 1993). World Sci. Publ., River Edge, NJ, p 259–267 Kudryavtseva and Mazorchuk [2008] Kudryavtseva G, Mazorchuk V (2008) Schur-Weyl dualities for symmetric inverse semigroups. J Pure Appl Algebra 212(8):1987–1995. 10.1016/j.jpaa.2007.12.004, URL https://doi.org/10.1016/j.jpaa.2007.12.004 Lawson [1998] Lawson MV (1998) Inverse semigroups. World Scientific Publishing Co. Inc., River Edge, NJ, 10.1142/9789812816689, URL http://dx.doi.org/10.1142/9789812816689, the theory of partial symmetries Martin [1994] Martin P (1994) Temperley-Lieb algebras for nonplanar statistical mechanics—the partition algebra construction. J Knot Theory Ramifications 3(1):51–82. 10.1142/S0218216594000071, URL http://dx.doi.org/10.1142/S0218216594000071 Martin [1996] Martin P (1996) The structure of the partition algebras. J Algebra 183(2):319–358. 10.1006/jabr.1996.0223, URL http://dx.doi.org/10.1006/jabr.1996.0223 Martin and Mazorchuk [2014] Martin P, Mazorchuk V (2014) On the representation theory of partial Brauer algebras. Q J Math 65(1):225–247. 10.1093/qmath/has043, URL https://doi.org/10.1093/qmath/has043 Martin [1990] Martin PP (1990) Representations of graph Temperley-Lieb algebras. Publ Res Inst Math Sci 26(3):485–503. 10.2977/prims/1195170958, URL https://doi.org/10.2977/prims/1195170958 Martin [2008] Martin S (2008) Schur algebras and representation theory, Cambridge Tracts in Mathematics, vol 112. Cambridge University Press, Cambridge, reprint of the 1993 original Munn [1955] Munn WD (1955) On semigroup algebras. Proc Cambridge Philos Soc 51:1–15 Munn [1957a] Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Dipper R, Doty S, Hu J (2008) Brauer algebras, symplectic Schur algebras and Schur-Weyl duality. Trans Amer Math Soc 360(1):189–213. 10.1090/S0002-9947-07-04179-7, URL https://doi.org/10.1090/S0002-9947-07-04179-7 Doty and Hu [2009] Doty S, Hu J (2009) Schur-Weyl duality for orthogonal groups. Proc Lond Math Soc (3) 98(3):679–713. 10.1112/plms/pdn044, URL https://doi.org/10.1112/plms/pdn044 Goodman and Wallach [2009] Goodman R, Wallach NR (2009) Symmetry, representations, and invariants, Graduate Texts in Mathematics, vol 255. Springer, Dordrecht, 10.1007/978-0-387-79852-3, URL https://doi.org/10.1007/978-0-387-79852-3 Graham and Lehrer [1996] Graham JJ, Lehrer GI (1996) Cellular algebras. Invent Math 123(1):1–34. 10.1007/BF01232365, URL http://dx.doi.org/10.1007/BF01232365 Green [1980] Green JA (1980) Polynomial representations of GLnsubscriptGL𝑛{\rm GL}_{n}roman_GL start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT, Lecture Notes in Mathematics, vol 830. Springer-Verlag, Berlin Grood [2006] Grood C (2006) The rook partition algebra. J Combin Theory Ser A 113(2):325–351. 10.1016/j.jcta.2005.03.006, URL https://doi.org/10.1016/j.jcta.2005.03.006 Halverson [1996] Halverson T (1996) Characters of the centralizer algebras of mixed tensor representations of Gl(r,ℂ)Gl𝑟ℂ{\rm Gl}(r,\mathbb{C})roman_Gl ( italic_r , blackboard_C ) and the quantum group 𝔘q(ℊl(r,ℂ))subscript𝔘𝑞ℊ𝑙𝑟ℂ\mathfrak{U}_{q}({\mathcal{g}l}(r,\mathbb{C}))fraktur_U start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ( caligraphic_g italic_l ( italic_r , blackboard_C ) ). Pacific J Math 174(2):359–410. URL http://projecteuclid.org/euclid.pjm/1102365176 Halverson [2004] Halverson T (2004) Representations of the q𝑞qitalic_q-rook monoid. J Algebra 273(1):227–251. 10.1016/j.jalgebra.2003.11.002, URL http://dx.doi.org/10.1016/j.jalgebra.2003.11.002 Halverson and delMas [2014] Halverson T, delMas E (2014) Representations of the Rook-Brauer algebra. Comm Algebra 42(1):423–443. 10.1080/00927872.2012.716120, URL https://doi.org/10.1080/00927872.2012.716120 Halverson and Ram [2005] Halverson T, Ram A (2005) Partition algebras. European J Combin 26(6):869–921. 10.1016/j.ejc.2004.06.005, URL http://dx.doi.org/10.1016/j.ejc.2004.06.005 Jacobson [1989] Jacobson N (1989) Basic algebra. II, 2nd edn. W. H. Freeman and Company, New York Jones [1994] Jones VFR (1994) The Potts model and the symmetric group. In: Subfactors (Kyuzeso, 1993). World Sci. Publ., River Edge, NJ, p 259–267 Kudryavtseva and Mazorchuk [2008] Kudryavtseva G, Mazorchuk V (2008) Schur-Weyl dualities for symmetric inverse semigroups. J Pure Appl Algebra 212(8):1987–1995. 10.1016/j.jpaa.2007.12.004, URL https://doi.org/10.1016/j.jpaa.2007.12.004 Lawson [1998] Lawson MV (1998) Inverse semigroups. World Scientific Publishing Co. Inc., River Edge, NJ, 10.1142/9789812816689, URL http://dx.doi.org/10.1142/9789812816689, the theory of partial symmetries Martin [1994] Martin P (1994) Temperley-Lieb algebras for nonplanar statistical mechanics—the partition algebra construction. J Knot Theory Ramifications 3(1):51–82. 10.1142/S0218216594000071, URL http://dx.doi.org/10.1142/S0218216594000071 Martin [1996] Martin P (1996) The structure of the partition algebras. J Algebra 183(2):319–358. 10.1006/jabr.1996.0223, URL http://dx.doi.org/10.1006/jabr.1996.0223 Martin and Mazorchuk [2014] Martin P, Mazorchuk V (2014) On the representation theory of partial Brauer algebras. Q J Math 65(1):225–247. 10.1093/qmath/has043, URL https://doi.org/10.1093/qmath/has043 Martin [1990] Martin PP (1990) Representations of graph Temperley-Lieb algebras. Publ Res Inst Math Sci 26(3):485–503. 10.2977/prims/1195170958, URL https://doi.org/10.2977/prims/1195170958 Martin [2008] Martin S (2008) Schur algebras and representation theory, Cambridge Tracts in Mathematics, vol 112. Cambridge University Press, Cambridge, reprint of the 1993 original Munn [1955] Munn WD (1955) On semigroup algebras. Proc Cambridge Philos Soc 51:1–15 Munn [1957a] Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Doty S, Hu J (2009) Schur-Weyl duality for orthogonal groups. Proc Lond Math Soc (3) 98(3):679–713. 10.1112/plms/pdn044, URL https://doi.org/10.1112/plms/pdn044 Goodman and Wallach [2009] Goodman R, Wallach NR (2009) Symmetry, representations, and invariants, Graduate Texts in Mathematics, vol 255. Springer, Dordrecht, 10.1007/978-0-387-79852-3, URL https://doi.org/10.1007/978-0-387-79852-3 Graham and Lehrer [1996] Graham JJ, Lehrer GI (1996) Cellular algebras. Invent Math 123(1):1–34. 10.1007/BF01232365, URL http://dx.doi.org/10.1007/BF01232365 Green [1980] Green JA (1980) Polynomial representations of GLnsubscriptGL𝑛{\rm GL}_{n}roman_GL start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT, Lecture Notes in Mathematics, vol 830. Springer-Verlag, Berlin Grood [2006] Grood C (2006) The rook partition algebra. J Combin Theory Ser A 113(2):325–351. 10.1016/j.jcta.2005.03.006, URL https://doi.org/10.1016/j.jcta.2005.03.006 Halverson [1996] Halverson T (1996) Characters of the centralizer algebras of mixed tensor representations of Gl(r,ℂ)Gl𝑟ℂ{\rm Gl}(r,\mathbb{C})roman_Gl ( italic_r , blackboard_C ) and the quantum group 𝔘q(ℊl(r,ℂ))subscript𝔘𝑞ℊ𝑙𝑟ℂ\mathfrak{U}_{q}({\mathcal{g}l}(r,\mathbb{C}))fraktur_U start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ( caligraphic_g italic_l ( italic_r , blackboard_C ) ). Pacific J Math 174(2):359–410. URL http://projecteuclid.org/euclid.pjm/1102365176 Halverson [2004] Halverson T (2004) Representations of the q𝑞qitalic_q-rook monoid. J Algebra 273(1):227–251. 10.1016/j.jalgebra.2003.11.002, URL http://dx.doi.org/10.1016/j.jalgebra.2003.11.002 Halverson and delMas [2014] Halverson T, delMas E (2014) Representations of the Rook-Brauer algebra. Comm Algebra 42(1):423–443. 10.1080/00927872.2012.716120, URL https://doi.org/10.1080/00927872.2012.716120 Halverson and Ram [2005] Halverson T, Ram A (2005) Partition algebras. European J Combin 26(6):869–921. 10.1016/j.ejc.2004.06.005, URL http://dx.doi.org/10.1016/j.ejc.2004.06.005 Jacobson [1989] Jacobson N (1989) Basic algebra. II, 2nd edn. W. H. Freeman and Company, New York Jones [1994] Jones VFR (1994) The Potts model and the symmetric group. In: Subfactors (Kyuzeso, 1993). World Sci. Publ., River Edge, NJ, p 259–267 Kudryavtseva and Mazorchuk [2008] Kudryavtseva G, Mazorchuk V (2008) Schur-Weyl dualities for symmetric inverse semigroups. J Pure Appl Algebra 212(8):1987–1995. 10.1016/j.jpaa.2007.12.004, URL https://doi.org/10.1016/j.jpaa.2007.12.004 Lawson [1998] Lawson MV (1998) Inverse semigroups. World Scientific Publishing Co. Inc., River Edge, NJ, 10.1142/9789812816689, URL http://dx.doi.org/10.1142/9789812816689, the theory of partial symmetries Martin [1994] Martin P (1994) Temperley-Lieb algebras for nonplanar statistical mechanics—the partition algebra construction. J Knot Theory Ramifications 3(1):51–82. 10.1142/S0218216594000071, URL http://dx.doi.org/10.1142/S0218216594000071 Martin [1996] Martin P (1996) The structure of the partition algebras. J Algebra 183(2):319–358. 10.1006/jabr.1996.0223, URL http://dx.doi.org/10.1006/jabr.1996.0223 Martin and Mazorchuk [2014] Martin P, Mazorchuk V (2014) On the representation theory of partial Brauer algebras. Q J Math 65(1):225–247. 10.1093/qmath/has043, URL https://doi.org/10.1093/qmath/has043 Martin [1990] Martin PP (1990) Representations of graph Temperley-Lieb algebras. Publ Res Inst Math Sci 26(3):485–503. 10.2977/prims/1195170958, URL https://doi.org/10.2977/prims/1195170958 Martin [2008] Martin S (2008) Schur algebras and representation theory, Cambridge Tracts in Mathematics, vol 112. Cambridge University Press, Cambridge, reprint of the 1993 original Munn [1955] Munn WD (1955) On semigroup algebras. Proc Cambridge Philos Soc 51:1–15 Munn [1957a] Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Goodman R, Wallach NR (2009) Symmetry, representations, and invariants, Graduate Texts in Mathematics, vol 255. Springer, Dordrecht, 10.1007/978-0-387-79852-3, URL https://doi.org/10.1007/978-0-387-79852-3 Graham and Lehrer [1996] Graham JJ, Lehrer GI (1996) Cellular algebras. Invent Math 123(1):1–34. 10.1007/BF01232365, URL http://dx.doi.org/10.1007/BF01232365 Green [1980] Green JA (1980) Polynomial representations of GLnsubscriptGL𝑛{\rm GL}_{n}roman_GL start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT, Lecture Notes in Mathematics, vol 830. Springer-Verlag, Berlin Grood [2006] Grood C (2006) The rook partition algebra. J Combin Theory Ser A 113(2):325–351. 10.1016/j.jcta.2005.03.006, URL https://doi.org/10.1016/j.jcta.2005.03.006 Halverson [1996] Halverson T (1996) Characters of the centralizer algebras of mixed tensor representations of Gl(r,ℂ)Gl𝑟ℂ{\rm Gl}(r,\mathbb{C})roman_Gl ( italic_r , blackboard_C ) and the quantum group 𝔘q(ℊl(r,ℂ))subscript𝔘𝑞ℊ𝑙𝑟ℂ\mathfrak{U}_{q}({\mathcal{g}l}(r,\mathbb{C}))fraktur_U start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ( caligraphic_g italic_l ( italic_r , blackboard_C ) ). Pacific J Math 174(2):359–410. URL http://projecteuclid.org/euclid.pjm/1102365176 Halverson [2004] Halverson T (2004) Representations of the q𝑞qitalic_q-rook monoid. J Algebra 273(1):227–251. 10.1016/j.jalgebra.2003.11.002, URL http://dx.doi.org/10.1016/j.jalgebra.2003.11.002 Halverson and delMas [2014] Halverson T, delMas E (2014) Representations of the Rook-Brauer algebra. Comm Algebra 42(1):423–443. 10.1080/00927872.2012.716120, URL https://doi.org/10.1080/00927872.2012.716120 Halverson and Ram [2005] Halverson T, Ram A (2005) Partition algebras. European J Combin 26(6):869–921. 10.1016/j.ejc.2004.06.005, URL http://dx.doi.org/10.1016/j.ejc.2004.06.005 Jacobson [1989] Jacobson N (1989) Basic algebra. II, 2nd edn. W. H. Freeman and Company, New York Jones [1994] Jones VFR (1994) The Potts model and the symmetric group. In: Subfactors (Kyuzeso, 1993). World Sci. Publ., River Edge, NJ, p 259–267 Kudryavtseva and Mazorchuk [2008] Kudryavtseva G, Mazorchuk V (2008) Schur-Weyl dualities for symmetric inverse semigroups. J Pure Appl Algebra 212(8):1987–1995. 10.1016/j.jpaa.2007.12.004, URL https://doi.org/10.1016/j.jpaa.2007.12.004 Lawson [1998] Lawson MV (1998) Inverse semigroups. World Scientific Publishing Co. Inc., River Edge, NJ, 10.1142/9789812816689, URL http://dx.doi.org/10.1142/9789812816689, the theory of partial symmetries Martin [1994] Martin P (1994) Temperley-Lieb algebras for nonplanar statistical mechanics—the partition algebra construction. J Knot Theory Ramifications 3(1):51–82. 10.1142/S0218216594000071, URL http://dx.doi.org/10.1142/S0218216594000071 Martin [1996] Martin P (1996) The structure of the partition algebras. J Algebra 183(2):319–358. 10.1006/jabr.1996.0223, URL http://dx.doi.org/10.1006/jabr.1996.0223 Martin and Mazorchuk [2014] Martin P, Mazorchuk V (2014) On the representation theory of partial Brauer algebras. Q J Math 65(1):225–247. 10.1093/qmath/has043, URL https://doi.org/10.1093/qmath/has043 Martin [1990] Martin PP (1990) Representations of graph Temperley-Lieb algebras. Publ Res Inst Math Sci 26(3):485–503. 10.2977/prims/1195170958, URL https://doi.org/10.2977/prims/1195170958 Martin [2008] Martin S (2008) Schur algebras and representation theory, Cambridge Tracts in Mathematics, vol 112. Cambridge University Press, Cambridge, reprint of the 1993 original Munn [1955] Munn WD (1955) On semigroup algebras. Proc Cambridge Philos Soc 51:1–15 Munn [1957a] Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Graham JJ, Lehrer GI (1996) Cellular algebras. Invent Math 123(1):1–34. 10.1007/BF01232365, URL http://dx.doi.org/10.1007/BF01232365 Green [1980] Green JA (1980) Polynomial representations of GLnsubscriptGL𝑛{\rm GL}_{n}roman_GL start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT, Lecture Notes in Mathematics, vol 830. Springer-Verlag, Berlin Grood [2006] Grood C (2006) The rook partition algebra. J Combin Theory Ser A 113(2):325–351. 10.1016/j.jcta.2005.03.006, URL https://doi.org/10.1016/j.jcta.2005.03.006 Halverson [1996] Halverson T (1996) Characters of the centralizer algebras of mixed tensor representations of Gl(r,ℂ)Gl𝑟ℂ{\rm Gl}(r,\mathbb{C})roman_Gl ( italic_r , blackboard_C ) and the quantum group 𝔘q(ℊl(r,ℂ))subscript𝔘𝑞ℊ𝑙𝑟ℂ\mathfrak{U}_{q}({\mathcal{g}l}(r,\mathbb{C}))fraktur_U start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ( caligraphic_g italic_l ( italic_r , blackboard_C ) ). Pacific J Math 174(2):359–410. URL http://projecteuclid.org/euclid.pjm/1102365176 Halverson [2004] Halverson T (2004) Representations of the q𝑞qitalic_q-rook monoid. J Algebra 273(1):227–251. 10.1016/j.jalgebra.2003.11.002, URL http://dx.doi.org/10.1016/j.jalgebra.2003.11.002 Halverson and delMas [2014] Halverson T, delMas E (2014) Representations of the Rook-Brauer algebra. Comm Algebra 42(1):423–443. 10.1080/00927872.2012.716120, URL https://doi.org/10.1080/00927872.2012.716120 Halverson and Ram [2005] Halverson T, Ram A (2005) Partition algebras. European J Combin 26(6):869–921. 10.1016/j.ejc.2004.06.005, URL http://dx.doi.org/10.1016/j.ejc.2004.06.005 Jacobson [1989] Jacobson N (1989) Basic algebra. II, 2nd edn. W. H. Freeman and Company, New York Jones [1994] Jones VFR (1994) The Potts model and the symmetric group. In: Subfactors (Kyuzeso, 1993). World Sci. Publ., River Edge, NJ, p 259–267 Kudryavtseva and Mazorchuk [2008] Kudryavtseva G, Mazorchuk V (2008) Schur-Weyl dualities for symmetric inverse semigroups. J Pure Appl Algebra 212(8):1987–1995. 10.1016/j.jpaa.2007.12.004, URL https://doi.org/10.1016/j.jpaa.2007.12.004 Lawson [1998] Lawson MV (1998) Inverse semigroups. World Scientific Publishing Co. Inc., River Edge, NJ, 10.1142/9789812816689, URL http://dx.doi.org/10.1142/9789812816689, the theory of partial symmetries Martin [1994] Martin P (1994) Temperley-Lieb algebras for nonplanar statistical mechanics—the partition algebra construction. J Knot Theory Ramifications 3(1):51–82. 10.1142/S0218216594000071, URL http://dx.doi.org/10.1142/S0218216594000071 Martin [1996] Martin P (1996) The structure of the partition algebras. J Algebra 183(2):319–358. 10.1006/jabr.1996.0223, URL http://dx.doi.org/10.1006/jabr.1996.0223 Martin and Mazorchuk [2014] Martin P, Mazorchuk V (2014) On the representation theory of partial Brauer algebras. Q J Math 65(1):225–247. 10.1093/qmath/has043, URL https://doi.org/10.1093/qmath/has043 Martin [1990] Martin PP (1990) Representations of graph Temperley-Lieb algebras. Publ Res Inst Math Sci 26(3):485–503. 10.2977/prims/1195170958, URL https://doi.org/10.2977/prims/1195170958 Martin [2008] Martin S (2008) Schur algebras and representation theory, Cambridge Tracts in Mathematics, vol 112. Cambridge University Press, Cambridge, reprint of the 1993 original Munn [1955] Munn WD (1955) On semigroup algebras. Proc Cambridge Philos Soc 51:1–15 Munn [1957a] Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Green JA (1980) Polynomial representations of GLnsubscriptGL𝑛{\rm GL}_{n}roman_GL start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT, Lecture Notes in Mathematics, vol 830. Springer-Verlag, Berlin Grood [2006] Grood C (2006) The rook partition algebra. J Combin Theory Ser A 113(2):325–351. 10.1016/j.jcta.2005.03.006, URL https://doi.org/10.1016/j.jcta.2005.03.006 Halverson [1996] Halverson T (1996) Characters of the centralizer algebras of mixed tensor representations of Gl(r,ℂ)Gl𝑟ℂ{\rm Gl}(r,\mathbb{C})roman_Gl ( italic_r , blackboard_C ) and the quantum group 𝔘q(ℊl(r,ℂ))subscript𝔘𝑞ℊ𝑙𝑟ℂ\mathfrak{U}_{q}({\mathcal{g}l}(r,\mathbb{C}))fraktur_U start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ( caligraphic_g italic_l ( italic_r , blackboard_C ) ). Pacific J Math 174(2):359–410. URL http://projecteuclid.org/euclid.pjm/1102365176 Halverson [2004] Halverson T (2004) Representations of the q𝑞qitalic_q-rook monoid. J Algebra 273(1):227–251. 10.1016/j.jalgebra.2003.11.002, URL http://dx.doi.org/10.1016/j.jalgebra.2003.11.002 Halverson and delMas [2014] Halverson T, delMas E (2014) Representations of the Rook-Brauer algebra. Comm Algebra 42(1):423–443. 10.1080/00927872.2012.716120, URL https://doi.org/10.1080/00927872.2012.716120 Halverson and Ram [2005] Halverson T, Ram A (2005) Partition algebras. European J Combin 26(6):869–921. 10.1016/j.ejc.2004.06.005, URL http://dx.doi.org/10.1016/j.ejc.2004.06.005 Jacobson [1989] Jacobson N (1989) Basic algebra. II, 2nd edn. W. H. Freeman and Company, New York Jones [1994] Jones VFR (1994) The Potts model and the symmetric group. In: Subfactors (Kyuzeso, 1993). World Sci. Publ., River Edge, NJ, p 259–267 Kudryavtseva and Mazorchuk [2008] Kudryavtseva G, Mazorchuk V (2008) Schur-Weyl dualities for symmetric inverse semigroups. J Pure Appl Algebra 212(8):1987–1995. 10.1016/j.jpaa.2007.12.004, URL https://doi.org/10.1016/j.jpaa.2007.12.004 Lawson [1998] Lawson MV (1998) Inverse semigroups. World Scientific Publishing Co. Inc., River Edge, NJ, 10.1142/9789812816689, URL http://dx.doi.org/10.1142/9789812816689, the theory of partial symmetries Martin [1994] Martin P (1994) Temperley-Lieb algebras for nonplanar statistical mechanics—the partition algebra construction. J Knot Theory Ramifications 3(1):51–82. 10.1142/S0218216594000071, URL http://dx.doi.org/10.1142/S0218216594000071 Martin [1996] Martin P (1996) The structure of the partition algebras. J Algebra 183(2):319–358. 10.1006/jabr.1996.0223, URL http://dx.doi.org/10.1006/jabr.1996.0223 Martin and Mazorchuk [2014] Martin P, Mazorchuk V (2014) On the representation theory of partial Brauer algebras. Q J Math 65(1):225–247. 10.1093/qmath/has043, URL https://doi.org/10.1093/qmath/has043 Martin [1990] Martin PP (1990) Representations of graph Temperley-Lieb algebras. Publ Res Inst Math Sci 26(3):485–503. 10.2977/prims/1195170958, URL https://doi.org/10.2977/prims/1195170958 Martin [2008] Martin S (2008) Schur algebras and representation theory, Cambridge Tracts in Mathematics, vol 112. Cambridge University Press, Cambridge, reprint of the 1993 original Munn [1955] Munn WD (1955) On semigroup algebras. Proc Cambridge Philos Soc 51:1–15 Munn [1957a] Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Grood C (2006) The rook partition algebra. J Combin Theory Ser A 113(2):325–351. 10.1016/j.jcta.2005.03.006, URL https://doi.org/10.1016/j.jcta.2005.03.006 Halverson [1996] Halverson T (1996) Characters of the centralizer algebras of mixed tensor representations of Gl(r,ℂ)Gl𝑟ℂ{\rm Gl}(r,\mathbb{C})roman_Gl ( italic_r , blackboard_C ) and the quantum group 𝔘q(ℊl(r,ℂ))subscript𝔘𝑞ℊ𝑙𝑟ℂ\mathfrak{U}_{q}({\mathcal{g}l}(r,\mathbb{C}))fraktur_U start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ( caligraphic_g italic_l ( italic_r , blackboard_C ) ). Pacific J Math 174(2):359–410. URL http://projecteuclid.org/euclid.pjm/1102365176 Halverson [2004] Halverson T (2004) Representations of the q𝑞qitalic_q-rook monoid. J Algebra 273(1):227–251. 10.1016/j.jalgebra.2003.11.002, URL http://dx.doi.org/10.1016/j.jalgebra.2003.11.002 Halverson and delMas [2014] Halverson T, delMas E (2014) Representations of the Rook-Brauer algebra. Comm Algebra 42(1):423–443. 10.1080/00927872.2012.716120, URL https://doi.org/10.1080/00927872.2012.716120 Halverson and Ram [2005] Halverson T, Ram A (2005) Partition algebras. European J Combin 26(6):869–921. 10.1016/j.ejc.2004.06.005, URL http://dx.doi.org/10.1016/j.ejc.2004.06.005 Jacobson [1989] Jacobson N (1989) Basic algebra. II, 2nd edn. W. H. Freeman and Company, New York Jones [1994] Jones VFR (1994) The Potts model and the symmetric group. In: Subfactors (Kyuzeso, 1993). World Sci. Publ., River Edge, NJ, p 259–267 Kudryavtseva and Mazorchuk [2008] Kudryavtseva G, Mazorchuk V (2008) Schur-Weyl dualities for symmetric inverse semigroups. J Pure Appl Algebra 212(8):1987–1995. 10.1016/j.jpaa.2007.12.004, URL https://doi.org/10.1016/j.jpaa.2007.12.004 Lawson [1998] Lawson MV (1998) Inverse semigroups. World Scientific Publishing Co. Inc., River Edge, NJ, 10.1142/9789812816689, URL http://dx.doi.org/10.1142/9789812816689, the theory of partial symmetries Martin [1994] Martin P (1994) Temperley-Lieb algebras for nonplanar statistical mechanics—the partition algebra construction. J Knot Theory Ramifications 3(1):51–82. 10.1142/S0218216594000071, URL http://dx.doi.org/10.1142/S0218216594000071 Martin [1996] Martin P (1996) The structure of the partition algebras. J Algebra 183(2):319–358. 10.1006/jabr.1996.0223, URL http://dx.doi.org/10.1006/jabr.1996.0223 Martin and Mazorchuk [2014] Martin P, Mazorchuk V (2014) On the representation theory of partial Brauer algebras. Q J Math 65(1):225–247. 10.1093/qmath/has043, URL https://doi.org/10.1093/qmath/has043 Martin [1990] Martin PP (1990) Representations of graph Temperley-Lieb algebras. Publ Res Inst Math Sci 26(3):485–503. 10.2977/prims/1195170958, URL https://doi.org/10.2977/prims/1195170958 Martin [2008] Martin S (2008) Schur algebras and representation theory, Cambridge Tracts in Mathematics, vol 112. Cambridge University Press, Cambridge, reprint of the 1993 original Munn [1955] Munn WD (1955) On semigroup algebras. Proc Cambridge Philos Soc 51:1–15 Munn [1957a] Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Halverson T (1996) Characters of the centralizer algebras of mixed tensor representations of Gl(r,ℂ)Gl𝑟ℂ{\rm Gl}(r,\mathbb{C})roman_Gl ( italic_r , blackboard_C ) and the quantum group 𝔘q(ℊl(r,ℂ))subscript𝔘𝑞ℊ𝑙𝑟ℂ\mathfrak{U}_{q}({\mathcal{g}l}(r,\mathbb{C}))fraktur_U start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ( caligraphic_g italic_l ( italic_r , blackboard_C ) ). Pacific J Math 174(2):359–410. URL http://projecteuclid.org/euclid.pjm/1102365176 Halverson [2004] Halverson T (2004) Representations of the q𝑞qitalic_q-rook monoid. J Algebra 273(1):227–251. 10.1016/j.jalgebra.2003.11.002, URL http://dx.doi.org/10.1016/j.jalgebra.2003.11.002 Halverson and delMas [2014] Halverson T, delMas E (2014) Representations of the Rook-Brauer algebra. Comm Algebra 42(1):423–443. 10.1080/00927872.2012.716120, URL https://doi.org/10.1080/00927872.2012.716120 Halverson and Ram [2005] Halverson T, Ram A (2005) Partition algebras. European J Combin 26(6):869–921. 10.1016/j.ejc.2004.06.005, URL http://dx.doi.org/10.1016/j.ejc.2004.06.005 Jacobson [1989] Jacobson N (1989) Basic algebra. II, 2nd edn. W. H. Freeman and Company, New York Jones [1994] Jones VFR (1994) The Potts model and the symmetric group. In: Subfactors (Kyuzeso, 1993). World Sci. Publ., River Edge, NJ, p 259–267 Kudryavtseva and Mazorchuk [2008] Kudryavtseva G, Mazorchuk V (2008) Schur-Weyl dualities for symmetric inverse semigroups. J Pure Appl Algebra 212(8):1987–1995. 10.1016/j.jpaa.2007.12.004, URL https://doi.org/10.1016/j.jpaa.2007.12.004 Lawson [1998] Lawson MV (1998) Inverse semigroups. World Scientific Publishing Co. Inc., River Edge, NJ, 10.1142/9789812816689, URL http://dx.doi.org/10.1142/9789812816689, the theory of partial symmetries Martin [1994] Martin P (1994) Temperley-Lieb algebras for nonplanar statistical mechanics—the partition algebra construction. J Knot Theory Ramifications 3(1):51–82. 10.1142/S0218216594000071, URL http://dx.doi.org/10.1142/S0218216594000071 Martin [1996] Martin P (1996) The structure of the partition algebras. J Algebra 183(2):319–358. 10.1006/jabr.1996.0223, URL http://dx.doi.org/10.1006/jabr.1996.0223 Martin and Mazorchuk [2014] Martin P, Mazorchuk V (2014) On the representation theory of partial Brauer algebras. Q J Math 65(1):225–247. 10.1093/qmath/has043, URL https://doi.org/10.1093/qmath/has043 Martin [1990] Martin PP (1990) Representations of graph Temperley-Lieb algebras. Publ Res Inst Math Sci 26(3):485–503. 10.2977/prims/1195170958, URL https://doi.org/10.2977/prims/1195170958 Martin [2008] Martin S (2008) Schur algebras and representation theory, Cambridge Tracts in Mathematics, vol 112. Cambridge University Press, Cambridge, reprint of the 1993 original Munn [1955] Munn WD (1955) On semigroup algebras. Proc Cambridge Philos Soc 51:1–15 Munn [1957a] Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Halverson T (2004) Representations of the q𝑞qitalic_q-rook monoid. J Algebra 273(1):227–251. 10.1016/j.jalgebra.2003.11.002, URL http://dx.doi.org/10.1016/j.jalgebra.2003.11.002 Halverson and delMas [2014] Halverson T, delMas E (2014) Representations of the Rook-Brauer algebra. Comm Algebra 42(1):423–443. 10.1080/00927872.2012.716120, URL https://doi.org/10.1080/00927872.2012.716120 Halverson and Ram [2005] Halverson T, Ram A (2005) Partition algebras. European J Combin 26(6):869–921. 10.1016/j.ejc.2004.06.005, URL http://dx.doi.org/10.1016/j.ejc.2004.06.005 Jacobson [1989] Jacobson N (1989) Basic algebra. II, 2nd edn. W. H. Freeman and Company, New York Jones [1994] Jones VFR (1994) The Potts model and the symmetric group. In: Subfactors (Kyuzeso, 1993). World Sci. Publ., River Edge, NJ, p 259–267 Kudryavtseva and Mazorchuk [2008] Kudryavtseva G, Mazorchuk V (2008) Schur-Weyl dualities for symmetric inverse semigroups. J Pure Appl Algebra 212(8):1987–1995. 10.1016/j.jpaa.2007.12.004, URL https://doi.org/10.1016/j.jpaa.2007.12.004 Lawson [1998] Lawson MV (1998) Inverse semigroups. World Scientific Publishing Co. Inc., River Edge, NJ, 10.1142/9789812816689, URL http://dx.doi.org/10.1142/9789812816689, the theory of partial symmetries Martin [1994] Martin P (1994) Temperley-Lieb algebras for nonplanar statistical mechanics—the partition algebra construction. J Knot Theory Ramifications 3(1):51–82. 10.1142/S0218216594000071, URL http://dx.doi.org/10.1142/S0218216594000071 Martin [1996] Martin P (1996) The structure of the partition algebras. J Algebra 183(2):319–358. 10.1006/jabr.1996.0223, URL http://dx.doi.org/10.1006/jabr.1996.0223 Martin and Mazorchuk [2014] Martin P, Mazorchuk V (2014) On the representation theory of partial Brauer algebras. Q J Math 65(1):225–247. 10.1093/qmath/has043, URL https://doi.org/10.1093/qmath/has043 Martin [1990] Martin PP (1990) Representations of graph Temperley-Lieb algebras. Publ Res Inst Math Sci 26(3):485–503. 10.2977/prims/1195170958, URL https://doi.org/10.2977/prims/1195170958 Martin [2008] Martin S (2008) Schur algebras and representation theory, Cambridge Tracts in Mathematics, vol 112. Cambridge University Press, Cambridge, reprint of the 1993 original Munn [1955] Munn WD (1955) On semigroup algebras. Proc Cambridge Philos Soc 51:1–15 Munn [1957a] Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Halverson T, delMas E (2014) Representations of the Rook-Brauer algebra. Comm Algebra 42(1):423–443. 10.1080/00927872.2012.716120, URL https://doi.org/10.1080/00927872.2012.716120 Halverson and Ram [2005] Halverson T, Ram A (2005) Partition algebras. European J Combin 26(6):869–921. 10.1016/j.ejc.2004.06.005, URL http://dx.doi.org/10.1016/j.ejc.2004.06.005 Jacobson [1989] Jacobson N (1989) Basic algebra. II, 2nd edn. W. H. Freeman and Company, New York Jones [1994] Jones VFR (1994) The Potts model and the symmetric group. In: Subfactors (Kyuzeso, 1993). World Sci. Publ., River Edge, NJ, p 259–267 Kudryavtseva and Mazorchuk [2008] Kudryavtseva G, Mazorchuk V (2008) Schur-Weyl dualities for symmetric inverse semigroups. J Pure Appl Algebra 212(8):1987–1995. 10.1016/j.jpaa.2007.12.004, URL https://doi.org/10.1016/j.jpaa.2007.12.004 Lawson [1998] Lawson MV (1998) Inverse semigroups. World Scientific Publishing Co. Inc., River Edge, NJ, 10.1142/9789812816689, URL http://dx.doi.org/10.1142/9789812816689, the theory of partial symmetries Martin [1994] Martin P (1994) Temperley-Lieb algebras for nonplanar statistical mechanics—the partition algebra construction. J Knot Theory Ramifications 3(1):51–82. 10.1142/S0218216594000071, URL http://dx.doi.org/10.1142/S0218216594000071 Martin [1996] Martin P (1996) The structure of the partition algebras. J Algebra 183(2):319–358. 10.1006/jabr.1996.0223, URL http://dx.doi.org/10.1006/jabr.1996.0223 Martin and Mazorchuk [2014] Martin P, Mazorchuk V (2014) On the representation theory of partial Brauer algebras. Q J Math 65(1):225–247. 10.1093/qmath/has043, URL https://doi.org/10.1093/qmath/has043 Martin [1990] Martin PP (1990) Representations of graph Temperley-Lieb algebras. Publ Res Inst Math Sci 26(3):485–503. 10.2977/prims/1195170958, URL https://doi.org/10.2977/prims/1195170958 Martin [2008] Martin S (2008) Schur algebras and representation theory, Cambridge Tracts in Mathematics, vol 112. Cambridge University Press, Cambridge, reprint of the 1993 original Munn [1955] Munn WD (1955) On semigroup algebras. Proc Cambridge Philos Soc 51:1–15 Munn [1957a] Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Halverson T, Ram A (2005) Partition algebras. European J Combin 26(6):869–921. 10.1016/j.ejc.2004.06.005, URL http://dx.doi.org/10.1016/j.ejc.2004.06.005 Jacobson [1989] Jacobson N (1989) Basic algebra. II, 2nd edn. W. H. Freeman and Company, New York Jones [1994] Jones VFR (1994) The Potts model and the symmetric group. In: Subfactors (Kyuzeso, 1993). World Sci. Publ., River Edge, NJ, p 259–267 Kudryavtseva and Mazorchuk [2008] Kudryavtseva G, Mazorchuk V (2008) Schur-Weyl dualities for symmetric inverse semigroups. J Pure Appl Algebra 212(8):1987–1995. 10.1016/j.jpaa.2007.12.004, URL https://doi.org/10.1016/j.jpaa.2007.12.004 Lawson [1998] Lawson MV (1998) Inverse semigroups. World Scientific Publishing Co. Inc., River Edge, NJ, 10.1142/9789812816689, URL http://dx.doi.org/10.1142/9789812816689, the theory of partial symmetries Martin [1994] Martin P (1994) Temperley-Lieb algebras for nonplanar statistical mechanics—the partition algebra construction. J Knot Theory Ramifications 3(1):51–82. 10.1142/S0218216594000071, URL http://dx.doi.org/10.1142/S0218216594000071 Martin [1996] Martin P (1996) The structure of the partition algebras. J Algebra 183(2):319–358. 10.1006/jabr.1996.0223, URL http://dx.doi.org/10.1006/jabr.1996.0223 Martin and Mazorchuk [2014] Martin P, Mazorchuk V (2014) On the representation theory of partial Brauer algebras. Q J Math 65(1):225–247. 10.1093/qmath/has043, URL https://doi.org/10.1093/qmath/has043 Martin [1990] Martin PP (1990) Representations of graph Temperley-Lieb algebras. Publ Res Inst Math Sci 26(3):485–503. 10.2977/prims/1195170958, URL https://doi.org/10.2977/prims/1195170958 Martin [2008] Martin S (2008) Schur algebras and representation theory, Cambridge Tracts in Mathematics, vol 112. Cambridge University Press, Cambridge, reprint of the 1993 original Munn [1955] Munn WD (1955) On semigroup algebras. Proc Cambridge Philos Soc 51:1–15 Munn [1957a] Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Jacobson N (1989) Basic algebra. II, 2nd edn. W. H. Freeman and Company, New York Jones [1994] Jones VFR (1994) The Potts model and the symmetric group. In: Subfactors (Kyuzeso, 1993). World Sci. Publ., River Edge, NJ, p 259–267 Kudryavtseva and Mazorchuk [2008] Kudryavtseva G, Mazorchuk V (2008) Schur-Weyl dualities for symmetric inverse semigroups. J Pure Appl Algebra 212(8):1987–1995. 10.1016/j.jpaa.2007.12.004, URL https://doi.org/10.1016/j.jpaa.2007.12.004 Lawson [1998] Lawson MV (1998) Inverse semigroups. World Scientific Publishing Co. Inc., River Edge, NJ, 10.1142/9789812816689, URL http://dx.doi.org/10.1142/9789812816689, the theory of partial symmetries Martin [1994] Martin P (1994) Temperley-Lieb algebras for nonplanar statistical mechanics—the partition algebra construction. J Knot Theory Ramifications 3(1):51–82. 10.1142/S0218216594000071, URL http://dx.doi.org/10.1142/S0218216594000071 Martin [1996] Martin P (1996) The structure of the partition algebras. J Algebra 183(2):319–358. 10.1006/jabr.1996.0223, URL http://dx.doi.org/10.1006/jabr.1996.0223 Martin and Mazorchuk [2014] Martin P, Mazorchuk V (2014) On the representation theory of partial Brauer algebras. Q J Math 65(1):225–247. 10.1093/qmath/has043, URL https://doi.org/10.1093/qmath/has043 Martin [1990] Martin PP (1990) Representations of graph Temperley-Lieb algebras. Publ Res Inst Math Sci 26(3):485–503. 10.2977/prims/1195170958, URL https://doi.org/10.2977/prims/1195170958 Martin [2008] Martin S (2008) Schur algebras and representation theory, Cambridge Tracts in Mathematics, vol 112. Cambridge University Press, Cambridge, reprint of the 1993 original Munn [1955] Munn WD (1955) On semigroup algebras. Proc Cambridge Philos Soc 51:1–15 Munn [1957a] Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Jones VFR (1994) The Potts model and the symmetric group. In: Subfactors (Kyuzeso, 1993). World Sci. Publ., River Edge, NJ, p 259–267 Kudryavtseva and Mazorchuk [2008] Kudryavtseva G, Mazorchuk V (2008) Schur-Weyl dualities for symmetric inverse semigroups. J Pure Appl Algebra 212(8):1987–1995. 10.1016/j.jpaa.2007.12.004, URL https://doi.org/10.1016/j.jpaa.2007.12.004 Lawson [1998] Lawson MV (1998) Inverse semigroups. World Scientific Publishing Co. Inc., River Edge, NJ, 10.1142/9789812816689, URL http://dx.doi.org/10.1142/9789812816689, the theory of partial symmetries Martin [1994] Martin P (1994) Temperley-Lieb algebras for nonplanar statistical mechanics—the partition algebra construction. J Knot Theory Ramifications 3(1):51–82. 10.1142/S0218216594000071, URL http://dx.doi.org/10.1142/S0218216594000071 Martin [1996] Martin P (1996) The structure of the partition algebras. J Algebra 183(2):319–358. 10.1006/jabr.1996.0223, URL http://dx.doi.org/10.1006/jabr.1996.0223 Martin and Mazorchuk [2014] Martin P, Mazorchuk V (2014) On the representation theory of partial Brauer algebras. Q J Math 65(1):225–247. 10.1093/qmath/has043, URL https://doi.org/10.1093/qmath/has043 Martin [1990] Martin PP (1990) Representations of graph Temperley-Lieb algebras. Publ Res Inst Math Sci 26(3):485–503. 10.2977/prims/1195170958, URL https://doi.org/10.2977/prims/1195170958 Martin [2008] Martin S (2008) Schur algebras and representation theory, Cambridge Tracts in Mathematics, vol 112. Cambridge University Press, Cambridge, reprint of the 1993 original Munn [1955] Munn WD (1955) On semigroup algebras. Proc Cambridge Philos Soc 51:1–15 Munn [1957a] Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Kudryavtseva G, Mazorchuk V (2008) Schur-Weyl dualities for symmetric inverse semigroups. J Pure Appl Algebra 212(8):1987–1995. 10.1016/j.jpaa.2007.12.004, URL https://doi.org/10.1016/j.jpaa.2007.12.004 Lawson [1998] Lawson MV (1998) Inverse semigroups. World Scientific Publishing Co. Inc., River Edge, NJ, 10.1142/9789812816689, URL http://dx.doi.org/10.1142/9789812816689, the theory of partial symmetries Martin [1994] Martin P (1994) Temperley-Lieb algebras for nonplanar statistical mechanics—the partition algebra construction. J Knot Theory Ramifications 3(1):51–82. 10.1142/S0218216594000071, URL http://dx.doi.org/10.1142/S0218216594000071 Martin [1996] Martin P (1996) The structure of the partition algebras. J Algebra 183(2):319–358. 10.1006/jabr.1996.0223, URL http://dx.doi.org/10.1006/jabr.1996.0223 Martin and Mazorchuk [2014] Martin P, Mazorchuk V (2014) On the representation theory of partial Brauer algebras. Q J Math 65(1):225–247. 10.1093/qmath/has043, URL https://doi.org/10.1093/qmath/has043 Martin [1990] Martin PP (1990) Representations of graph Temperley-Lieb algebras. Publ Res Inst Math Sci 26(3):485–503. 10.2977/prims/1195170958, URL https://doi.org/10.2977/prims/1195170958 Martin [2008] Martin S (2008) Schur algebras and representation theory, Cambridge Tracts in Mathematics, vol 112. Cambridge University Press, Cambridge, reprint of the 1993 original Munn [1955] Munn WD (1955) On semigroup algebras. Proc Cambridge Philos Soc 51:1–15 Munn [1957a] Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Lawson MV (1998) Inverse semigroups. World Scientific Publishing Co. Inc., River Edge, NJ, 10.1142/9789812816689, URL http://dx.doi.org/10.1142/9789812816689, the theory of partial symmetries Martin [1994] Martin P (1994) Temperley-Lieb algebras for nonplanar statistical mechanics—the partition algebra construction. J Knot Theory Ramifications 3(1):51–82. 10.1142/S0218216594000071, URL http://dx.doi.org/10.1142/S0218216594000071 Martin [1996] Martin P (1996) The structure of the partition algebras. J Algebra 183(2):319–358. 10.1006/jabr.1996.0223, URL http://dx.doi.org/10.1006/jabr.1996.0223 Martin and Mazorchuk [2014] Martin P, Mazorchuk V (2014) On the representation theory of partial Brauer algebras. Q J Math 65(1):225–247. 10.1093/qmath/has043, URL https://doi.org/10.1093/qmath/has043 Martin [1990] Martin PP (1990) Representations of graph Temperley-Lieb algebras. Publ Res Inst Math Sci 26(3):485–503. 10.2977/prims/1195170958, URL https://doi.org/10.2977/prims/1195170958 Martin [2008] Martin S (2008) Schur algebras and representation theory, Cambridge Tracts in Mathematics, vol 112. Cambridge University Press, Cambridge, reprint of the 1993 original Munn [1955] Munn WD (1955) On semigroup algebras. Proc Cambridge Philos Soc 51:1–15 Munn [1957a] Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Martin P (1994) Temperley-Lieb algebras for nonplanar statistical mechanics—the partition algebra construction. J Knot Theory Ramifications 3(1):51–82. 10.1142/S0218216594000071, URL http://dx.doi.org/10.1142/S0218216594000071 Martin [1996] Martin P (1996) The structure of the partition algebras. J Algebra 183(2):319–358. 10.1006/jabr.1996.0223, URL http://dx.doi.org/10.1006/jabr.1996.0223 Martin and Mazorchuk [2014] Martin P, Mazorchuk V (2014) On the representation theory of partial Brauer algebras. Q J Math 65(1):225–247. 10.1093/qmath/has043, URL https://doi.org/10.1093/qmath/has043 Martin [1990] Martin PP (1990) Representations of graph Temperley-Lieb algebras. Publ Res Inst Math Sci 26(3):485–503. 10.2977/prims/1195170958, URL https://doi.org/10.2977/prims/1195170958 Martin [2008] Martin S (2008) Schur algebras and representation theory, Cambridge Tracts in Mathematics, vol 112. Cambridge University Press, Cambridge, reprint of the 1993 original Munn [1955] Munn WD (1955) On semigroup algebras. Proc Cambridge Philos Soc 51:1–15 Munn [1957a] Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Martin P (1996) The structure of the partition algebras. J Algebra 183(2):319–358. 10.1006/jabr.1996.0223, URL http://dx.doi.org/10.1006/jabr.1996.0223 Martin and Mazorchuk [2014] Martin P, Mazorchuk V (2014) On the representation theory of partial Brauer algebras. Q J Math 65(1):225–247. 10.1093/qmath/has043, URL https://doi.org/10.1093/qmath/has043 Martin [1990] Martin PP (1990) Representations of graph Temperley-Lieb algebras. Publ Res Inst Math Sci 26(3):485–503. 10.2977/prims/1195170958, URL https://doi.org/10.2977/prims/1195170958 Martin [2008] Martin S (2008) Schur algebras and representation theory, Cambridge Tracts in Mathematics, vol 112. Cambridge University Press, Cambridge, reprint of the 1993 original Munn [1955] Munn WD (1955) On semigroup algebras. Proc Cambridge Philos Soc 51:1–15 Munn [1957a] Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Martin P, Mazorchuk V (2014) On the representation theory of partial Brauer algebras. Q J Math 65(1):225–247. 10.1093/qmath/has043, URL https://doi.org/10.1093/qmath/has043 Martin [1990] Martin PP (1990) Representations of graph Temperley-Lieb algebras. Publ Res Inst Math Sci 26(3):485–503. 10.2977/prims/1195170958, URL https://doi.org/10.2977/prims/1195170958 Martin [2008] Martin S (2008) Schur algebras and representation theory, Cambridge Tracts in Mathematics, vol 112. Cambridge University Press, Cambridge, reprint of the 1993 original Munn [1955] Munn WD (1955) On semigroup algebras. Proc Cambridge Philos Soc 51:1–15 Munn [1957a] Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Martin PP (1990) Representations of graph Temperley-Lieb algebras. Publ Res Inst Math Sci 26(3):485–503. 10.2977/prims/1195170958, URL https://doi.org/10.2977/prims/1195170958 Martin [2008] Martin S (2008) Schur algebras and representation theory, Cambridge Tracts in Mathematics, vol 112. Cambridge University Press, Cambridge, reprint of the 1993 original Munn [1955] Munn WD (1955) On semigroup algebras. Proc Cambridge Philos Soc 51:1–15 Munn [1957a] Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Martin S (2008) Schur algebras and representation theory, Cambridge Tracts in Mathematics, vol 112. Cambridge University Press, Cambridge, reprint of the 1993 original Munn [1955] Munn WD (1955) On semigroup algebras. Proc Cambridge Philos Soc 51:1–15 Munn [1957a] Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Munn WD (1955) On semigroup algebras. Proc Cambridge Philos Soc 51:1–15 Munn [1957a] Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. 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J Combin Theory Ser A 113(2):325–351. 10.1016/j.jcta.2005.03.006, URL https://doi.org/10.1016/j.jcta.2005.03.006 Halverson [1996] Halverson T (1996) Characters of the centralizer algebras of mixed tensor representations of Gl(r,ℂ)Gl𝑟ℂ{\rm Gl}(r,\mathbb{C})roman_Gl ( italic_r , blackboard_C ) and the quantum group 𝔘q(ℊl(r,ℂ))subscript𝔘𝑞ℊ𝑙𝑟ℂ\mathfrak{U}_{q}({\mathcal{g}l}(r,\mathbb{C}))fraktur_U start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ( caligraphic_g italic_l ( italic_r , blackboard_C ) ). Pacific J Math 174(2):359–410. URL http://projecteuclid.org/euclid.pjm/1102365176 Halverson [2004] Halverson T (2004) Representations of the q𝑞qitalic_q-rook monoid. J Algebra 273(1):227–251. 10.1016/j.jalgebra.2003.11.002, URL http://dx.doi.org/10.1016/j.jalgebra.2003.11.002 Halverson and delMas [2014] Halverson T, delMas E (2014) Representations of the Rook-Brauer algebra. Comm Algebra 42(1):423–443. 10.1080/00927872.2012.716120, URL https://doi.org/10.1080/00927872.2012.716120 Halverson and Ram [2005] Halverson T, Ram A (2005) Partition algebras. European J Combin 26(6):869–921. 10.1016/j.ejc.2004.06.005, URL http://dx.doi.org/10.1016/j.ejc.2004.06.005 Jacobson [1989] Jacobson N (1989) Basic algebra. II, 2nd edn. W. H. Freeman and Company, New York Jones [1994] Jones VFR (1994) The Potts model and the symmetric group. In: Subfactors (Kyuzeso, 1993). World Sci. Publ., River Edge, NJ, p 259–267 Kudryavtseva and Mazorchuk [2008] Kudryavtseva G, Mazorchuk V (2008) Schur-Weyl dualities for symmetric inverse semigroups. J Pure Appl Algebra 212(8):1987–1995. 10.1016/j.jpaa.2007.12.004, URL https://doi.org/10.1016/j.jpaa.2007.12.004 Lawson [1998] Lawson MV (1998) Inverse semigroups. World Scientific Publishing Co. Inc., River Edge, NJ, 10.1142/9789812816689, URL http://dx.doi.org/10.1142/9789812816689, the theory of partial symmetries Martin [1994] Martin P (1994) Temperley-Lieb algebras for nonplanar statistical mechanics—the partition algebra construction. J Knot Theory Ramifications 3(1):51–82. 10.1142/S0218216594000071, URL http://dx.doi.org/10.1142/S0218216594000071 Martin [1996] Martin P (1996) The structure of the partition algebras. J Algebra 183(2):319–358. 10.1006/jabr.1996.0223, URL http://dx.doi.org/10.1006/jabr.1996.0223 Martin and Mazorchuk [2014] Martin P, Mazorchuk V (2014) On the representation theory of partial Brauer algebras. Q J Math 65(1):225–247. 10.1093/qmath/has043, URL https://doi.org/10.1093/qmath/has043 Martin [1990] Martin PP (1990) Representations of graph Temperley-Lieb algebras. Publ Res Inst Math Sci 26(3):485–503. 10.2977/prims/1195170958, URL https://doi.org/10.2977/prims/1195170958 Martin [2008] Martin S (2008) Schur algebras and representation theory, Cambridge Tracts in Mathematics, vol 112. Cambridge University Press, Cambridge, reprint of the 1993 original Munn [1955] Munn WD (1955) On semigroup algebras. Proc Cambridge Philos Soc 51:1–15 Munn [1957a] Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. de Concini C, Procesi C (1976) A characteristic free approach to invariant theory. Advances in Math 21(3):330–354 Curtis and Reiner [1981] Curtis CW, Reiner I (1981) Methods of representation theory. Vol. I. John Wiley & Sons Inc., New York, with applications to finite groups and orders, Pure and Applied Mathematics, A Wiley-Interscience Publication Dipper et al [2008] Dipper R, Doty S, Hu J (2008) Brauer algebras, symplectic Schur algebras and Schur-Weyl duality. Trans Amer Math Soc 360(1):189–213. 10.1090/S0002-9947-07-04179-7, URL https://doi.org/10.1090/S0002-9947-07-04179-7 Doty and Hu [2009] Doty S, Hu J (2009) Schur-Weyl duality for orthogonal groups. Proc Lond Math Soc (3) 98(3):679–713. 10.1112/plms/pdn044, URL https://doi.org/10.1112/plms/pdn044 Goodman and Wallach [2009] Goodman R, Wallach NR (2009) Symmetry, representations, and invariants, Graduate Texts in Mathematics, vol 255. Springer, Dordrecht, 10.1007/978-0-387-79852-3, URL https://doi.org/10.1007/978-0-387-79852-3 Graham and Lehrer [1996] Graham JJ, Lehrer GI (1996) Cellular algebras. Invent Math 123(1):1–34. 10.1007/BF01232365, URL http://dx.doi.org/10.1007/BF01232365 Green [1980] Green JA (1980) Polynomial representations of GLnsubscriptGL𝑛{\rm GL}_{n}roman_GL start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT, Lecture Notes in Mathematics, vol 830. Springer-Verlag, Berlin Grood [2006] Grood C (2006) The rook partition algebra. J Combin Theory Ser A 113(2):325–351. 10.1016/j.jcta.2005.03.006, URL https://doi.org/10.1016/j.jcta.2005.03.006 Halverson [1996] Halverson T (1996) Characters of the centralizer algebras of mixed tensor representations of Gl(r,ℂ)Gl𝑟ℂ{\rm Gl}(r,\mathbb{C})roman_Gl ( italic_r , blackboard_C ) and the quantum group 𝔘q(ℊl(r,ℂ))subscript𝔘𝑞ℊ𝑙𝑟ℂ\mathfrak{U}_{q}({\mathcal{g}l}(r,\mathbb{C}))fraktur_U start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ( caligraphic_g italic_l ( italic_r , blackboard_C ) ). Pacific J Math 174(2):359–410. URL http://projecteuclid.org/euclid.pjm/1102365176 Halverson [2004] Halverson T (2004) Representations of the q𝑞qitalic_q-rook monoid. J Algebra 273(1):227–251. 10.1016/j.jalgebra.2003.11.002, URL http://dx.doi.org/10.1016/j.jalgebra.2003.11.002 Halverson and delMas [2014] Halverson T, delMas E (2014) Representations of the Rook-Brauer algebra. Comm Algebra 42(1):423–443. 10.1080/00927872.2012.716120, URL https://doi.org/10.1080/00927872.2012.716120 Halverson and Ram [2005] Halverson T, Ram A (2005) Partition algebras. European J Combin 26(6):869–921. 10.1016/j.ejc.2004.06.005, URL http://dx.doi.org/10.1016/j.ejc.2004.06.005 Jacobson [1989] Jacobson N (1989) Basic algebra. II, 2nd edn. W. H. Freeman and Company, New York Jones [1994] Jones VFR (1994) The Potts model and the symmetric group. In: Subfactors (Kyuzeso, 1993). World Sci. Publ., River Edge, NJ, p 259–267 Kudryavtseva and Mazorchuk [2008] Kudryavtseva G, Mazorchuk V (2008) Schur-Weyl dualities for symmetric inverse semigroups. J Pure Appl Algebra 212(8):1987–1995. 10.1016/j.jpaa.2007.12.004, URL https://doi.org/10.1016/j.jpaa.2007.12.004 Lawson [1998] Lawson MV (1998) Inverse semigroups. World Scientific Publishing Co. Inc., River Edge, NJ, 10.1142/9789812816689, URL http://dx.doi.org/10.1142/9789812816689, the theory of partial symmetries Martin [1994] Martin P (1994) Temperley-Lieb algebras for nonplanar statistical mechanics—the partition algebra construction. J Knot Theory Ramifications 3(1):51–82. 10.1142/S0218216594000071, URL http://dx.doi.org/10.1142/S0218216594000071 Martin [1996] Martin P (1996) The structure of the partition algebras. J Algebra 183(2):319–358. 10.1006/jabr.1996.0223, URL http://dx.doi.org/10.1006/jabr.1996.0223 Martin and Mazorchuk [2014] Martin P, Mazorchuk V (2014) On the representation theory of partial Brauer algebras. Q J Math 65(1):225–247. 10.1093/qmath/has043, URL https://doi.org/10.1093/qmath/has043 Martin [1990] Martin PP (1990) Representations of graph Temperley-Lieb algebras. Publ Res Inst Math Sci 26(3):485–503. 10.2977/prims/1195170958, URL https://doi.org/10.2977/prims/1195170958 Martin [2008] Martin S (2008) Schur algebras and representation theory, Cambridge Tracts in Mathematics, vol 112. Cambridge University Press, Cambridge, reprint of the 1993 original Munn [1955] Munn WD (1955) On semigroup algebras. Proc Cambridge Philos Soc 51:1–15 Munn [1957a] Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Curtis CW, Reiner I (1981) Methods of representation theory. Vol. I. John Wiley & Sons Inc., New York, with applications to finite groups and orders, Pure and Applied Mathematics, A Wiley-Interscience Publication Dipper et al [2008] Dipper R, Doty S, Hu J (2008) Brauer algebras, symplectic Schur algebras and Schur-Weyl duality. Trans Amer Math Soc 360(1):189–213. 10.1090/S0002-9947-07-04179-7, URL https://doi.org/10.1090/S0002-9947-07-04179-7 Doty and Hu [2009] Doty S, Hu J (2009) Schur-Weyl duality for orthogonal groups. Proc Lond Math Soc (3) 98(3):679–713. 10.1112/plms/pdn044, URL https://doi.org/10.1112/plms/pdn044 Goodman and Wallach [2009] Goodman R, Wallach NR (2009) Symmetry, representations, and invariants, Graduate Texts in Mathematics, vol 255. Springer, Dordrecht, 10.1007/978-0-387-79852-3, URL https://doi.org/10.1007/978-0-387-79852-3 Graham and Lehrer [1996] Graham JJ, Lehrer GI (1996) Cellular algebras. Invent Math 123(1):1–34. 10.1007/BF01232365, URL http://dx.doi.org/10.1007/BF01232365 Green [1980] Green JA (1980) Polynomial representations of GLnsubscriptGL𝑛{\rm GL}_{n}roman_GL start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT, Lecture Notes in Mathematics, vol 830. Springer-Verlag, Berlin Grood [2006] Grood C (2006) The rook partition algebra. J Combin Theory Ser A 113(2):325–351. 10.1016/j.jcta.2005.03.006, URL https://doi.org/10.1016/j.jcta.2005.03.006 Halverson [1996] Halverson T (1996) Characters of the centralizer algebras of mixed tensor representations of Gl(r,ℂ)Gl𝑟ℂ{\rm Gl}(r,\mathbb{C})roman_Gl ( italic_r , blackboard_C ) and the quantum group 𝔘q(ℊl(r,ℂ))subscript𝔘𝑞ℊ𝑙𝑟ℂ\mathfrak{U}_{q}({\mathcal{g}l}(r,\mathbb{C}))fraktur_U start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ( caligraphic_g italic_l ( italic_r , blackboard_C ) ). Pacific J Math 174(2):359–410. URL http://projecteuclid.org/euclid.pjm/1102365176 Halverson [2004] Halverson T (2004) Representations of the q𝑞qitalic_q-rook monoid. J Algebra 273(1):227–251. 10.1016/j.jalgebra.2003.11.002, URL http://dx.doi.org/10.1016/j.jalgebra.2003.11.002 Halverson and delMas [2014] Halverson T, delMas E (2014) Representations of the Rook-Brauer algebra. Comm Algebra 42(1):423–443. 10.1080/00927872.2012.716120, URL https://doi.org/10.1080/00927872.2012.716120 Halverson and Ram [2005] Halverson T, Ram A (2005) Partition algebras. European J Combin 26(6):869–921. 10.1016/j.ejc.2004.06.005, URL http://dx.doi.org/10.1016/j.ejc.2004.06.005 Jacobson [1989] Jacobson N (1989) Basic algebra. II, 2nd edn. W. H. Freeman and Company, New York Jones [1994] Jones VFR (1994) The Potts model and the symmetric group. In: Subfactors (Kyuzeso, 1993). World Sci. Publ., River Edge, NJ, p 259–267 Kudryavtseva and Mazorchuk [2008] Kudryavtseva G, Mazorchuk V (2008) Schur-Weyl dualities for symmetric inverse semigroups. J Pure Appl Algebra 212(8):1987–1995. 10.1016/j.jpaa.2007.12.004, URL https://doi.org/10.1016/j.jpaa.2007.12.004 Lawson [1998] Lawson MV (1998) Inverse semigroups. World Scientific Publishing Co. Inc., River Edge, NJ, 10.1142/9789812816689, URL http://dx.doi.org/10.1142/9789812816689, the theory of partial symmetries Martin [1994] Martin P (1994) Temperley-Lieb algebras for nonplanar statistical mechanics—the partition algebra construction. J Knot Theory Ramifications 3(1):51–82. 10.1142/S0218216594000071, URL http://dx.doi.org/10.1142/S0218216594000071 Martin [1996] Martin P (1996) The structure of the partition algebras. J Algebra 183(2):319–358. 10.1006/jabr.1996.0223, URL http://dx.doi.org/10.1006/jabr.1996.0223 Martin and Mazorchuk [2014] Martin P, Mazorchuk V (2014) On the representation theory of partial Brauer algebras. Q J Math 65(1):225–247. 10.1093/qmath/has043, URL https://doi.org/10.1093/qmath/has043 Martin [1990] Martin PP (1990) Representations of graph Temperley-Lieb algebras. Publ Res Inst Math Sci 26(3):485–503. 10.2977/prims/1195170958, URL https://doi.org/10.2977/prims/1195170958 Martin [2008] Martin S (2008) Schur algebras and representation theory, Cambridge Tracts in Mathematics, vol 112. Cambridge University Press, Cambridge, reprint of the 1993 original Munn [1955] Munn WD (1955) On semigroup algebras. Proc Cambridge Philos Soc 51:1–15 Munn [1957a] Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Dipper R, Doty S, Hu J (2008) Brauer algebras, symplectic Schur algebras and Schur-Weyl duality. Trans Amer Math Soc 360(1):189–213. 10.1090/S0002-9947-07-04179-7, URL https://doi.org/10.1090/S0002-9947-07-04179-7 Doty and Hu [2009] Doty S, Hu J (2009) Schur-Weyl duality for orthogonal groups. Proc Lond Math Soc (3) 98(3):679–713. 10.1112/plms/pdn044, URL https://doi.org/10.1112/plms/pdn044 Goodman and Wallach [2009] Goodman R, Wallach NR (2009) Symmetry, representations, and invariants, Graduate Texts in Mathematics, vol 255. Springer, Dordrecht, 10.1007/978-0-387-79852-3, URL https://doi.org/10.1007/978-0-387-79852-3 Graham and Lehrer [1996] Graham JJ, Lehrer GI (1996) Cellular algebras. Invent Math 123(1):1–34. 10.1007/BF01232365, URL http://dx.doi.org/10.1007/BF01232365 Green [1980] Green JA (1980) Polynomial representations of GLnsubscriptGL𝑛{\rm GL}_{n}roman_GL start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT, Lecture Notes in Mathematics, vol 830. Springer-Verlag, Berlin Grood [2006] Grood C (2006) The rook partition algebra. J Combin Theory Ser A 113(2):325–351. 10.1016/j.jcta.2005.03.006, URL https://doi.org/10.1016/j.jcta.2005.03.006 Halverson [1996] Halverson T (1996) Characters of the centralizer algebras of mixed tensor representations of Gl(r,ℂ)Gl𝑟ℂ{\rm Gl}(r,\mathbb{C})roman_Gl ( italic_r , blackboard_C ) and the quantum group 𝔘q(ℊl(r,ℂ))subscript𝔘𝑞ℊ𝑙𝑟ℂ\mathfrak{U}_{q}({\mathcal{g}l}(r,\mathbb{C}))fraktur_U start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ( caligraphic_g italic_l ( italic_r , blackboard_C ) ). Pacific J Math 174(2):359–410. URL http://projecteuclid.org/euclid.pjm/1102365176 Halverson [2004] Halverson T (2004) Representations of the q𝑞qitalic_q-rook monoid. J Algebra 273(1):227–251. 10.1016/j.jalgebra.2003.11.002, URL http://dx.doi.org/10.1016/j.jalgebra.2003.11.002 Halverson and delMas [2014] Halverson T, delMas E (2014) Representations of the Rook-Brauer algebra. Comm Algebra 42(1):423–443. 10.1080/00927872.2012.716120, URL https://doi.org/10.1080/00927872.2012.716120 Halverson and Ram [2005] Halverson T, Ram A (2005) Partition algebras. European J Combin 26(6):869–921. 10.1016/j.ejc.2004.06.005, URL http://dx.doi.org/10.1016/j.ejc.2004.06.005 Jacobson [1989] Jacobson N (1989) Basic algebra. II, 2nd edn. W. H. Freeman and Company, New York Jones [1994] Jones VFR (1994) The Potts model and the symmetric group. In: Subfactors (Kyuzeso, 1993). World Sci. Publ., River Edge, NJ, p 259–267 Kudryavtseva and Mazorchuk [2008] Kudryavtseva G, Mazorchuk V (2008) Schur-Weyl dualities for symmetric inverse semigroups. J Pure Appl Algebra 212(8):1987–1995. 10.1016/j.jpaa.2007.12.004, URL https://doi.org/10.1016/j.jpaa.2007.12.004 Lawson [1998] Lawson MV (1998) Inverse semigroups. World Scientific Publishing Co. Inc., River Edge, NJ, 10.1142/9789812816689, URL http://dx.doi.org/10.1142/9789812816689, the theory of partial symmetries Martin [1994] Martin P (1994) Temperley-Lieb algebras for nonplanar statistical mechanics—the partition algebra construction. J Knot Theory Ramifications 3(1):51–82. 10.1142/S0218216594000071, URL http://dx.doi.org/10.1142/S0218216594000071 Martin [1996] Martin P (1996) The structure of the partition algebras. J Algebra 183(2):319–358. 10.1006/jabr.1996.0223, URL http://dx.doi.org/10.1006/jabr.1996.0223 Martin and Mazorchuk [2014] Martin P, Mazorchuk V (2014) On the representation theory of partial Brauer algebras. Q J Math 65(1):225–247. 10.1093/qmath/has043, URL https://doi.org/10.1093/qmath/has043 Martin [1990] Martin PP (1990) Representations of graph Temperley-Lieb algebras. Publ Res Inst Math Sci 26(3):485–503. 10.2977/prims/1195170958, URL https://doi.org/10.2977/prims/1195170958 Martin [2008] Martin S (2008) Schur algebras and representation theory, Cambridge Tracts in Mathematics, vol 112. Cambridge University Press, Cambridge, reprint of the 1993 original Munn [1955] Munn WD (1955) On semigroup algebras. Proc Cambridge Philos Soc 51:1–15 Munn [1957a] Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Doty S, Hu J (2009) Schur-Weyl duality for orthogonal groups. Proc Lond Math Soc (3) 98(3):679–713. 10.1112/plms/pdn044, URL https://doi.org/10.1112/plms/pdn044 Goodman and Wallach [2009] Goodman R, Wallach NR (2009) Symmetry, representations, and invariants, Graduate Texts in Mathematics, vol 255. Springer, Dordrecht, 10.1007/978-0-387-79852-3, URL https://doi.org/10.1007/978-0-387-79852-3 Graham and Lehrer [1996] Graham JJ, Lehrer GI (1996) Cellular algebras. Invent Math 123(1):1–34. 10.1007/BF01232365, URL http://dx.doi.org/10.1007/BF01232365 Green [1980] Green JA (1980) Polynomial representations of GLnsubscriptGL𝑛{\rm GL}_{n}roman_GL start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT, Lecture Notes in Mathematics, vol 830. Springer-Verlag, Berlin Grood [2006] Grood C (2006) The rook partition algebra. J Combin Theory Ser A 113(2):325–351. 10.1016/j.jcta.2005.03.006, URL https://doi.org/10.1016/j.jcta.2005.03.006 Halverson [1996] Halverson T (1996) Characters of the centralizer algebras of mixed tensor representations of Gl(r,ℂ)Gl𝑟ℂ{\rm Gl}(r,\mathbb{C})roman_Gl ( italic_r , blackboard_C ) and the quantum group 𝔘q(ℊl(r,ℂ))subscript𝔘𝑞ℊ𝑙𝑟ℂ\mathfrak{U}_{q}({\mathcal{g}l}(r,\mathbb{C}))fraktur_U start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ( caligraphic_g italic_l ( italic_r , blackboard_C ) ). Pacific J Math 174(2):359–410. URL http://projecteuclid.org/euclid.pjm/1102365176 Halverson [2004] Halverson T (2004) Representations of the q𝑞qitalic_q-rook monoid. J Algebra 273(1):227–251. 10.1016/j.jalgebra.2003.11.002, URL http://dx.doi.org/10.1016/j.jalgebra.2003.11.002 Halverson and delMas [2014] Halverson T, delMas E (2014) Representations of the Rook-Brauer algebra. Comm Algebra 42(1):423–443. 10.1080/00927872.2012.716120, URL https://doi.org/10.1080/00927872.2012.716120 Halverson and Ram [2005] Halverson T, Ram A (2005) Partition algebras. European J Combin 26(6):869–921. 10.1016/j.ejc.2004.06.005, URL http://dx.doi.org/10.1016/j.ejc.2004.06.005 Jacobson [1989] Jacobson N (1989) Basic algebra. II, 2nd edn. W. H. Freeman and Company, New York Jones [1994] Jones VFR (1994) The Potts model and the symmetric group. In: Subfactors (Kyuzeso, 1993). World Sci. Publ., River Edge, NJ, p 259–267 Kudryavtseva and Mazorchuk [2008] Kudryavtseva G, Mazorchuk V (2008) Schur-Weyl dualities for symmetric inverse semigroups. J Pure Appl Algebra 212(8):1987–1995. 10.1016/j.jpaa.2007.12.004, URL https://doi.org/10.1016/j.jpaa.2007.12.004 Lawson [1998] Lawson MV (1998) Inverse semigroups. World Scientific Publishing Co. Inc., River Edge, NJ, 10.1142/9789812816689, URL http://dx.doi.org/10.1142/9789812816689, the theory of partial symmetries Martin [1994] Martin P (1994) Temperley-Lieb algebras for nonplanar statistical mechanics—the partition algebra construction. J Knot Theory Ramifications 3(1):51–82. 10.1142/S0218216594000071, URL http://dx.doi.org/10.1142/S0218216594000071 Martin [1996] Martin P (1996) The structure of the partition algebras. J Algebra 183(2):319–358. 10.1006/jabr.1996.0223, URL http://dx.doi.org/10.1006/jabr.1996.0223 Martin and Mazorchuk [2014] Martin P, Mazorchuk V (2014) On the representation theory of partial Brauer algebras. Q J Math 65(1):225–247. 10.1093/qmath/has043, URL https://doi.org/10.1093/qmath/has043 Martin [1990] Martin PP (1990) Representations of graph Temperley-Lieb algebras. Publ Res Inst Math Sci 26(3):485–503. 10.2977/prims/1195170958, URL https://doi.org/10.2977/prims/1195170958 Martin [2008] Martin S (2008) Schur algebras and representation theory, Cambridge Tracts in Mathematics, vol 112. Cambridge University Press, Cambridge, reprint of the 1993 original Munn [1955] Munn WD (1955) On semigroup algebras. Proc Cambridge Philos Soc 51:1–15 Munn [1957a] Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Goodman R, Wallach NR (2009) Symmetry, representations, and invariants, Graduate Texts in Mathematics, vol 255. Springer, Dordrecht, 10.1007/978-0-387-79852-3, URL https://doi.org/10.1007/978-0-387-79852-3 Graham and Lehrer [1996] Graham JJ, Lehrer GI (1996) Cellular algebras. Invent Math 123(1):1–34. 10.1007/BF01232365, URL http://dx.doi.org/10.1007/BF01232365 Green [1980] Green JA (1980) Polynomial representations of GLnsubscriptGL𝑛{\rm GL}_{n}roman_GL start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT, Lecture Notes in Mathematics, vol 830. Springer-Verlag, Berlin Grood [2006] Grood C (2006) The rook partition algebra. J Combin Theory Ser A 113(2):325–351. 10.1016/j.jcta.2005.03.006, URL https://doi.org/10.1016/j.jcta.2005.03.006 Halverson [1996] Halverson T (1996) Characters of the centralizer algebras of mixed tensor representations of Gl(r,ℂ)Gl𝑟ℂ{\rm Gl}(r,\mathbb{C})roman_Gl ( italic_r , blackboard_C ) and the quantum group 𝔘q(ℊl(r,ℂ))subscript𝔘𝑞ℊ𝑙𝑟ℂ\mathfrak{U}_{q}({\mathcal{g}l}(r,\mathbb{C}))fraktur_U start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ( caligraphic_g italic_l ( italic_r , blackboard_C ) ). Pacific J Math 174(2):359–410. URL http://projecteuclid.org/euclid.pjm/1102365176 Halverson [2004] Halverson T (2004) Representations of the q𝑞qitalic_q-rook monoid. J Algebra 273(1):227–251. 10.1016/j.jalgebra.2003.11.002, URL http://dx.doi.org/10.1016/j.jalgebra.2003.11.002 Halverson and delMas [2014] Halverson T, delMas E (2014) Representations of the Rook-Brauer algebra. Comm Algebra 42(1):423–443. 10.1080/00927872.2012.716120, URL https://doi.org/10.1080/00927872.2012.716120 Halverson and Ram [2005] Halverson T, Ram A (2005) Partition algebras. European J Combin 26(6):869–921. 10.1016/j.ejc.2004.06.005, URL http://dx.doi.org/10.1016/j.ejc.2004.06.005 Jacobson [1989] Jacobson N (1989) Basic algebra. II, 2nd edn. W. H. Freeman and Company, New York Jones [1994] Jones VFR (1994) The Potts model and the symmetric group. In: Subfactors (Kyuzeso, 1993). World Sci. Publ., River Edge, NJ, p 259–267 Kudryavtseva and Mazorchuk [2008] Kudryavtseva G, Mazorchuk V (2008) Schur-Weyl dualities for symmetric inverse semigroups. J Pure Appl Algebra 212(8):1987–1995. 10.1016/j.jpaa.2007.12.004, URL https://doi.org/10.1016/j.jpaa.2007.12.004 Lawson [1998] Lawson MV (1998) Inverse semigroups. World Scientific Publishing Co. Inc., River Edge, NJ, 10.1142/9789812816689, URL http://dx.doi.org/10.1142/9789812816689, the theory of partial symmetries Martin [1994] Martin P (1994) Temperley-Lieb algebras for nonplanar statistical mechanics—the partition algebra construction. J Knot Theory Ramifications 3(1):51–82. 10.1142/S0218216594000071, URL http://dx.doi.org/10.1142/S0218216594000071 Martin [1996] Martin P (1996) The structure of the partition algebras. J Algebra 183(2):319–358. 10.1006/jabr.1996.0223, URL http://dx.doi.org/10.1006/jabr.1996.0223 Martin and Mazorchuk [2014] Martin P, Mazorchuk V (2014) On the representation theory of partial Brauer algebras. Q J Math 65(1):225–247. 10.1093/qmath/has043, URL https://doi.org/10.1093/qmath/has043 Martin [1990] Martin PP (1990) Representations of graph Temperley-Lieb algebras. Publ Res Inst Math Sci 26(3):485–503. 10.2977/prims/1195170958, URL https://doi.org/10.2977/prims/1195170958 Martin [2008] Martin S (2008) Schur algebras and representation theory, Cambridge Tracts in Mathematics, vol 112. Cambridge University Press, Cambridge, reprint of the 1993 original Munn [1955] Munn WD (1955) On semigroup algebras. Proc Cambridge Philos Soc 51:1–15 Munn [1957a] Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Graham JJ, Lehrer GI (1996) Cellular algebras. Invent Math 123(1):1–34. 10.1007/BF01232365, URL http://dx.doi.org/10.1007/BF01232365 Green [1980] Green JA (1980) Polynomial representations of GLnsubscriptGL𝑛{\rm GL}_{n}roman_GL start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT, Lecture Notes in Mathematics, vol 830. Springer-Verlag, Berlin Grood [2006] Grood C (2006) The rook partition algebra. J Combin Theory Ser A 113(2):325–351. 10.1016/j.jcta.2005.03.006, URL https://doi.org/10.1016/j.jcta.2005.03.006 Halverson [1996] Halverson T (1996) Characters of the centralizer algebras of mixed tensor representations of Gl(r,ℂ)Gl𝑟ℂ{\rm Gl}(r,\mathbb{C})roman_Gl ( italic_r , blackboard_C ) and the quantum group 𝔘q(ℊl(r,ℂ))subscript𝔘𝑞ℊ𝑙𝑟ℂ\mathfrak{U}_{q}({\mathcal{g}l}(r,\mathbb{C}))fraktur_U start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ( caligraphic_g italic_l ( italic_r , blackboard_C ) ). Pacific J Math 174(2):359–410. URL http://projecteuclid.org/euclid.pjm/1102365176 Halverson [2004] Halverson T (2004) Representations of the q𝑞qitalic_q-rook monoid. J Algebra 273(1):227–251. 10.1016/j.jalgebra.2003.11.002, URL http://dx.doi.org/10.1016/j.jalgebra.2003.11.002 Halverson and delMas [2014] Halverson T, delMas E (2014) Representations of the Rook-Brauer algebra. Comm Algebra 42(1):423–443. 10.1080/00927872.2012.716120, URL https://doi.org/10.1080/00927872.2012.716120 Halverson and Ram [2005] Halverson T, Ram A (2005) Partition algebras. European J Combin 26(6):869–921. 10.1016/j.ejc.2004.06.005, URL http://dx.doi.org/10.1016/j.ejc.2004.06.005 Jacobson [1989] Jacobson N (1989) Basic algebra. II, 2nd edn. W. H. Freeman and Company, New York Jones [1994] Jones VFR (1994) The Potts model and the symmetric group. In: Subfactors (Kyuzeso, 1993). World Sci. Publ., River Edge, NJ, p 259–267 Kudryavtseva and Mazorchuk [2008] Kudryavtseva G, Mazorchuk V (2008) Schur-Weyl dualities for symmetric inverse semigroups. J Pure Appl Algebra 212(8):1987–1995. 10.1016/j.jpaa.2007.12.004, URL https://doi.org/10.1016/j.jpaa.2007.12.004 Lawson [1998] Lawson MV (1998) Inverse semigroups. World Scientific Publishing Co. Inc., River Edge, NJ, 10.1142/9789812816689, URL http://dx.doi.org/10.1142/9789812816689, the theory of partial symmetries Martin [1994] Martin P (1994) Temperley-Lieb algebras for nonplanar statistical mechanics—the partition algebra construction. J Knot Theory Ramifications 3(1):51–82. 10.1142/S0218216594000071, URL http://dx.doi.org/10.1142/S0218216594000071 Martin [1996] Martin P (1996) The structure of the partition algebras. J Algebra 183(2):319–358. 10.1006/jabr.1996.0223, URL http://dx.doi.org/10.1006/jabr.1996.0223 Martin and Mazorchuk [2014] Martin P, Mazorchuk V (2014) On the representation theory of partial Brauer algebras. Q J Math 65(1):225–247. 10.1093/qmath/has043, URL https://doi.org/10.1093/qmath/has043 Martin [1990] Martin PP (1990) Representations of graph Temperley-Lieb algebras. Publ Res Inst Math Sci 26(3):485–503. 10.2977/prims/1195170958, URL https://doi.org/10.2977/prims/1195170958 Martin [2008] Martin S (2008) Schur algebras and representation theory, Cambridge Tracts in Mathematics, vol 112. Cambridge University Press, Cambridge, reprint of the 1993 original Munn [1955] Munn WD (1955) On semigroup algebras. Proc Cambridge Philos Soc 51:1–15 Munn [1957a] Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Green JA (1980) Polynomial representations of GLnsubscriptGL𝑛{\rm GL}_{n}roman_GL start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT, Lecture Notes in Mathematics, vol 830. Springer-Verlag, Berlin Grood [2006] Grood C (2006) The rook partition algebra. J Combin Theory Ser A 113(2):325–351. 10.1016/j.jcta.2005.03.006, URL https://doi.org/10.1016/j.jcta.2005.03.006 Halverson [1996] Halverson T (1996) Characters of the centralizer algebras of mixed tensor representations of Gl(r,ℂ)Gl𝑟ℂ{\rm Gl}(r,\mathbb{C})roman_Gl ( italic_r , blackboard_C ) and the quantum group 𝔘q(ℊl(r,ℂ))subscript𝔘𝑞ℊ𝑙𝑟ℂ\mathfrak{U}_{q}({\mathcal{g}l}(r,\mathbb{C}))fraktur_U start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ( caligraphic_g italic_l ( italic_r , blackboard_C ) ). Pacific J Math 174(2):359–410. URL http://projecteuclid.org/euclid.pjm/1102365176 Halverson [2004] Halverson T (2004) Representations of the q𝑞qitalic_q-rook monoid. J Algebra 273(1):227–251. 10.1016/j.jalgebra.2003.11.002, URL http://dx.doi.org/10.1016/j.jalgebra.2003.11.002 Halverson and delMas [2014] Halverson T, delMas E (2014) Representations of the Rook-Brauer algebra. Comm Algebra 42(1):423–443. 10.1080/00927872.2012.716120, URL https://doi.org/10.1080/00927872.2012.716120 Halverson and Ram [2005] Halverson T, Ram A (2005) Partition algebras. European J Combin 26(6):869–921. 10.1016/j.ejc.2004.06.005, URL http://dx.doi.org/10.1016/j.ejc.2004.06.005 Jacobson [1989] Jacobson N (1989) Basic algebra. II, 2nd edn. W. H. Freeman and Company, New York Jones [1994] Jones VFR (1994) The Potts model and the symmetric group. In: Subfactors (Kyuzeso, 1993). World Sci. Publ., River Edge, NJ, p 259–267 Kudryavtseva and Mazorchuk [2008] Kudryavtseva G, Mazorchuk V (2008) Schur-Weyl dualities for symmetric inverse semigroups. J Pure Appl Algebra 212(8):1987–1995. 10.1016/j.jpaa.2007.12.004, URL https://doi.org/10.1016/j.jpaa.2007.12.004 Lawson [1998] Lawson MV (1998) Inverse semigroups. World Scientific Publishing Co. Inc., River Edge, NJ, 10.1142/9789812816689, URL http://dx.doi.org/10.1142/9789812816689, the theory of partial symmetries Martin [1994] Martin P (1994) Temperley-Lieb algebras for nonplanar statistical mechanics—the partition algebra construction. J Knot Theory Ramifications 3(1):51–82. 10.1142/S0218216594000071, URL http://dx.doi.org/10.1142/S0218216594000071 Martin [1996] Martin P (1996) The structure of the partition algebras. J Algebra 183(2):319–358. 10.1006/jabr.1996.0223, URL http://dx.doi.org/10.1006/jabr.1996.0223 Martin and Mazorchuk [2014] Martin P, Mazorchuk V (2014) On the representation theory of partial Brauer algebras. Q J Math 65(1):225–247. 10.1093/qmath/has043, URL https://doi.org/10.1093/qmath/has043 Martin [1990] Martin PP (1990) Representations of graph Temperley-Lieb algebras. Publ Res Inst Math Sci 26(3):485–503. 10.2977/prims/1195170958, URL https://doi.org/10.2977/prims/1195170958 Martin [2008] Martin S (2008) Schur algebras and representation theory, Cambridge Tracts in Mathematics, vol 112. Cambridge University Press, Cambridge, reprint of the 1993 original Munn [1955] Munn WD (1955) On semigroup algebras. Proc Cambridge Philos Soc 51:1–15 Munn [1957a] Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Grood C (2006) The rook partition algebra. J Combin Theory Ser A 113(2):325–351. 10.1016/j.jcta.2005.03.006, URL https://doi.org/10.1016/j.jcta.2005.03.006 Halverson [1996] Halverson T (1996) Characters of the centralizer algebras of mixed tensor representations of Gl(r,ℂ)Gl𝑟ℂ{\rm Gl}(r,\mathbb{C})roman_Gl ( italic_r , blackboard_C ) and the quantum group 𝔘q(ℊl(r,ℂ))subscript𝔘𝑞ℊ𝑙𝑟ℂ\mathfrak{U}_{q}({\mathcal{g}l}(r,\mathbb{C}))fraktur_U start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ( caligraphic_g italic_l ( italic_r , blackboard_C ) ). Pacific J Math 174(2):359–410. URL http://projecteuclid.org/euclid.pjm/1102365176 Halverson [2004] Halverson T (2004) Representations of the q𝑞qitalic_q-rook monoid. J Algebra 273(1):227–251. 10.1016/j.jalgebra.2003.11.002, URL http://dx.doi.org/10.1016/j.jalgebra.2003.11.002 Halverson and delMas [2014] Halverson T, delMas E (2014) Representations of the Rook-Brauer algebra. Comm Algebra 42(1):423–443. 10.1080/00927872.2012.716120, URL https://doi.org/10.1080/00927872.2012.716120 Halverson and Ram [2005] Halverson T, Ram A (2005) Partition algebras. European J Combin 26(6):869–921. 10.1016/j.ejc.2004.06.005, URL http://dx.doi.org/10.1016/j.ejc.2004.06.005 Jacobson [1989] Jacobson N (1989) Basic algebra. II, 2nd edn. W. H. Freeman and Company, New York Jones [1994] Jones VFR (1994) The Potts model and the symmetric group. In: Subfactors (Kyuzeso, 1993). World Sci. Publ., River Edge, NJ, p 259–267 Kudryavtseva and Mazorchuk [2008] Kudryavtseva G, Mazorchuk V (2008) Schur-Weyl dualities for symmetric inverse semigroups. J Pure Appl Algebra 212(8):1987–1995. 10.1016/j.jpaa.2007.12.004, URL https://doi.org/10.1016/j.jpaa.2007.12.004 Lawson [1998] Lawson MV (1998) Inverse semigroups. World Scientific Publishing Co. Inc., River Edge, NJ, 10.1142/9789812816689, URL http://dx.doi.org/10.1142/9789812816689, the theory of partial symmetries Martin [1994] Martin P (1994) Temperley-Lieb algebras for nonplanar statistical mechanics—the partition algebra construction. J Knot Theory Ramifications 3(1):51–82. 10.1142/S0218216594000071, URL http://dx.doi.org/10.1142/S0218216594000071 Martin [1996] Martin P (1996) The structure of the partition algebras. J Algebra 183(2):319–358. 10.1006/jabr.1996.0223, URL http://dx.doi.org/10.1006/jabr.1996.0223 Martin and Mazorchuk [2014] Martin P, Mazorchuk V (2014) On the representation theory of partial Brauer algebras. Q J Math 65(1):225–247. 10.1093/qmath/has043, URL https://doi.org/10.1093/qmath/has043 Martin [1990] Martin PP (1990) Representations of graph Temperley-Lieb algebras. Publ Res Inst Math Sci 26(3):485–503. 10.2977/prims/1195170958, URL https://doi.org/10.2977/prims/1195170958 Martin [2008] Martin S (2008) Schur algebras and representation theory, Cambridge Tracts in Mathematics, vol 112. Cambridge University Press, Cambridge, reprint of the 1993 original Munn [1955] Munn WD (1955) On semigroup algebras. Proc Cambridge Philos Soc 51:1–15 Munn [1957a] Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Halverson T (1996) Characters of the centralizer algebras of mixed tensor representations of Gl(r,ℂ)Gl𝑟ℂ{\rm Gl}(r,\mathbb{C})roman_Gl ( italic_r , blackboard_C ) and the quantum group 𝔘q(ℊl(r,ℂ))subscript𝔘𝑞ℊ𝑙𝑟ℂ\mathfrak{U}_{q}({\mathcal{g}l}(r,\mathbb{C}))fraktur_U start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ( caligraphic_g italic_l ( italic_r , blackboard_C ) ). Pacific J Math 174(2):359–410. URL http://projecteuclid.org/euclid.pjm/1102365176 Halverson [2004] Halverson T (2004) Representations of the q𝑞qitalic_q-rook monoid. J Algebra 273(1):227–251. 10.1016/j.jalgebra.2003.11.002, URL http://dx.doi.org/10.1016/j.jalgebra.2003.11.002 Halverson and delMas [2014] Halverson T, delMas E (2014) Representations of the Rook-Brauer algebra. Comm Algebra 42(1):423–443. 10.1080/00927872.2012.716120, URL https://doi.org/10.1080/00927872.2012.716120 Halverson and Ram [2005] Halverson T, Ram A (2005) Partition algebras. European J Combin 26(6):869–921. 10.1016/j.ejc.2004.06.005, URL http://dx.doi.org/10.1016/j.ejc.2004.06.005 Jacobson [1989] Jacobson N (1989) Basic algebra. II, 2nd edn. W. H. Freeman and Company, New York Jones [1994] Jones VFR (1994) The Potts model and the symmetric group. In: Subfactors (Kyuzeso, 1993). World Sci. Publ., River Edge, NJ, p 259–267 Kudryavtseva and Mazorchuk [2008] Kudryavtseva G, Mazorchuk V (2008) Schur-Weyl dualities for symmetric inverse semigroups. J Pure Appl Algebra 212(8):1987–1995. 10.1016/j.jpaa.2007.12.004, URL https://doi.org/10.1016/j.jpaa.2007.12.004 Lawson [1998] Lawson MV (1998) Inverse semigroups. World Scientific Publishing Co. Inc., River Edge, NJ, 10.1142/9789812816689, URL http://dx.doi.org/10.1142/9789812816689, the theory of partial symmetries Martin [1994] Martin P (1994) Temperley-Lieb algebras for nonplanar statistical mechanics—the partition algebra construction. J Knot Theory Ramifications 3(1):51–82. 10.1142/S0218216594000071, URL http://dx.doi.org/10.1142/S0218216594000071 Martin [1996] Martin P (1996) The structure of the partition algebras. J Algebra 183(2):319–358. 10.1006/jabr.1996.0223, URL http://dx.doi.org/10.1006/jabr.1996.0223 Martin and Mazorchuk [2014] Martin P, Mazorchuk V (2014) On the representation theory of partial Brauer algebras. Q J Math 65(1):225–247. 10.1093/qmath/has043, URL https://doi.org/10.1093/qmath/has043 Martin [1990] Martin PP (1990) Representations of graph Temperley-Lieb algebras. Publ Res Inst Math Sci 26(3):485–503. 10.2977/prims/1195170958, URL https://doi.org/10.2977/prims/1195170958 Martin [2008] Martin S (2008) Schur algebras and representation theory, Cambridge Tracts in Mathematics, vol 112. Cambridge University Press, Cambridge, reprint of the 1993 original Munn [1955] Munn WD (1955) On semigroup algebras. Proc Cambridge Philos Soc 51:1–15 Munn [1957a] Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Halverson T (2004) Representations of the q𝑞qitalic_q-rook monoid. J Algebra 273(1):227–251. 10.1016/j.jalgebra.2003.11.002, URL http://dx.doi.org/10.1016/j.jalgebra.2003.11.002 Halverson and delMas [2014] Halverson T, delMas E (2014) Representations of the Rook-Brauer algebra. Comm Algebra 42(1):423–443. 10.1080/00927872.2012.716120, URL https://doi.org/10.1080/00927872.2012.716120 Halverson and Ram [2005] Halverson T, Ram A (2005) Partition algebras. European J Combin 26(6):869–921. 10.1016/j.ejc.2004.06.005, URL http://dx.doi.org/10.1016/j.ejc.2004.06.005 Jacobson [1989] Jacobson N (1989) Basic algebra. II, 2nd edn. W. H. Freeman and Company, New York Jones [1994] Jones VFR (1994) The Potts model and the symmetric group. In: Subfactors (Kyuzeso, 1993). World Sci. Publ., River Edge, NJ, p 259–267 Kudryavtseva and Mazorchuk [2008] Kudryavtseva G, Mazorchuk V (2008) Schur-Weyl dualities for symmetric inverse semigroups. J Pure Appl Algebra 212(8):1987–1995. 10.1016/j.jpaa.2007.12.004, URL https://doi.org/10.1016/j.jpaa.2007.12.004 Lawson [1998] Lawson MV (1998) Inverse semigroups. World Scientific Publishing Co. Inc., River Edge, NJ, 10.1142/9789812816689, URL http://dx.doi.org/10.1142/9789812816689, the theory of partial symmetries Martin [1994] Martin P (1994) Temperley-Lieb algebras for nonplanar statistical mechanics—the partition algebra construction. J Knot Theory Ramifications 3(1):51–82. 10.1142/S0218216594000071, URL http://dx.doi.org/10.1142/S0218216594000071 Martin [1996] Martin P (1996) The structure of the partition algebras. J Algebra 183(2):319–358. 10.1006/jabr.1996.0223, URL http://dx.doi.org/10.1006/jabr.1996.0223 Martin and Mazorchuk [2014] Martin P, Mazorchuk V (2014) On the representation theory of partial Brauer algebras. Q J Math 65(1):225–247. 10.1093/qmath/has043, URL https://doi.org/10.1093/qmath/has043 Martin [1990] Martin PP (1990) Representations of graph Temperley-Lieb algebras. Publ Res Inst Math Sci 26(3):485–503. 10.2977/prims/1195170958, URL https://doi.org/10.2977/prims/1195170958 Martin [2008] Martin S (2008) Schur algebras and representation theory, Cambridge Tracts in Mathematics, vol 112. Cambridge University Press, Cambridge, reprint of the 1993 original Munn [1955] Munn WD (1955) On semigroup algebras. Proc Cambridge Philos Soc 51:1–15 Munn [1957a] Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Halverson T, delMas E (2014) Representations of the Rook-Brauer algebra. Comm Algebra 42(1):423–443. 10.1080/00927872.2012.716120, URL https://doi.org/10.1080/00927872.2012.716120 Halverson and Ram [2005] Halverson T, Ram A (2005) Partition algebras. European J Combin 26(6):869–921. 10.1016/j.ejc.2004.06.005, URL http://dx.doi.org/10.1016/j.ejc.2004.06.005 Jacobson [1989] Jacobson N (1989) Basic algebra. II, 2nd edn. W. H. Freeman and Company, New York Jones [1994] Jones VFR (1994) The Potts model and the symmetric group. In: Subfactors (Kyuzeso, 1993). World Sci. Publ., River Edge, NJ, p 259–267 Kudryavtseva and Mazorchuk [2008] Kudryavtseva G, Mazorchuk V (2008) Schur-Weyl dualities for symmetric inverse semigroups. J Pure Appl Algebra 212(8):1987–1995. 10.1016/j.jpaa.2007.12.004, URL https://doi.org/10.1016/j.jpaa.2007.12.004 Lawson [1998] Lawson MV (1998) Inverse semigroups. World Scientific Publishing Co. Inc., River Edge, NJ, 10.1142/9789812816689, URL http://dx.doi.org/10.1142/9789812816689, the theory of partial symmetries Martin [1994] Martin P (1994) Temperley-Lieb algebras for nonplanar statistical mechanics—the partition algebra construction. J Knot Theory Ramifications 3(1):51–82. 10.1142/S0218216594000071, URL http://dx.doi.org/10.1142/S0218216594000071 Martin [1996] Martin P (1996) The structure of the partition algebras. J Algebra 183(2):319–358. 10.1006/jabr.1996.0223, URL http://dx.doi.org/10.1006/jabr.1996.0223 Martin and Mazorchuk [2014] Martin P, Mazorchuk V (2014) On the representation theory of partial Brauer algebras. Q J Math 65(1):225–247. 10.1093/qmath/has043, URL https://doi.org/10.1093/qmath/has043 Martin [1990] Martin PP (1990) Representations of graph Temperley-Lieb algebras. Publ Res Inst Math Sci 26(3):485–503. 10.2977/prims/1195170958, URL https://doi.org/10.2977/prims/1195170958 Martin [2008] Martin S (2008) Schur algebras and representation theory, Cambridge Tracts in Mathematics, vol 112. Cambridge University Press, Cambridge, reprint of the 1993 original Munn [1955] Munn WD (1955) On semigroup algebras. Proc Cambridge Philos Soc 51:1–15 Munn [1957a] Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Halverson T, Ram A (2005) Partition algebras. European J Combin 26(6):869–921. 10.1016/j.ejc.2004.06.005, URL http://dx.doi.org/10.1016/j.ejc.2004.06.005 Jacobson [1989] Jacobson N (1989) Basic algebra. II, 2nd edn. W. H. Freeman and Company, New York Jones [1994] Jones VFR (1994) The Potts model and the symmetric group. In: Subfactors (Kyuzeso, 1993). World Sci. Publ., River Edge, NJ, p 259–267 Kudryavtseva and Mazorchuk [2008] Kudryavtseva G, Mazorchuk V (2008) Schur-Weyl dualities for symmetric inverse semigroups. J Pure Appl Algebra 212(8):1987–1995. 10.1016/j.jpaa.2007.12.004, URL https://doi.org/10.1016/j.jpaa.2007.12.004 Lawson [1998] Lawson MV (1998) Inverse semigroups. World Scientific Publishing Co. Inc., River Edge, NJ, 10.1142/9789812816689, URL http://dx.doi.org/10.1142/9789812816689, the theory of partial symmetries Martin [1994] Martin P (1994) Temperley-Lieb algebras for nonplanar statistical mechanics—the partition algebra construction. J Knot Theory Ramifications 3(1):51–82. 10.1142/S0218216594000071, URL http://dx.doi.org/10.1142/S0218216594000071 Martin [1996] Martin P (1996) The structure of the partition algebras. J Algebra 183(2):319–358. 10.1006/jabr.1996.0223, URL http://dx.doi.org/10.1006/jabr.1996.0223 Martin and Mazorchuk [2014] Martin P, Mazorchuk V (2014) On the representation theory of partial Brauer algebras. Q J Math 65(1):225–247. 10.1093/qmath/has043, URL https://doi.org/10.1093/qmath/has043 Martin [1990] Martin PP (1990) Representations of graph Temperley-Lieb algebras. Publ Res Inst Math Sci 26(3):485–503. 10.2977/prims/1195170958, URL https://doi.org/10.2977/prims/1195170958 Martin [2008] Martin S (2008) Schur algebras and representation theory, Cambridge Tracts in Mathematics, vol 112. Cambridge University Press, Cambridge, reprint of the 1993 original Munn [1955] Munn WD (1955) On semigroup algebras. Proc Cambridge Philos Soc 51:1–15 Munn [1957a] Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Jacobson N (1989) Basic algebra. II, 2nd edn. W. H. Freeman and Company, New York Jones [1994] Jones VFR (1994) The Potts model and the symmetric group. In: Subfactors (Kyuzeso, 1993). World Sci. Publ., River Edge, NJ, p 259–267 Kudryavtseva and Mazorchuk [2008] Kudryavtseva G, Mazorchuk V (2008) Schur-Weyl dualities for symmetric inverse semigroups. J Pure Appl Algebra 212(8):1987–1995. 10.1016/j.jpaa.2007.12.004, URL https://doi.org/10.1016/j.jpaa.2007.12.004 Lawson [1998] Lawson MV (1998) Inverse semigroups. World Scientific Publishing Co. Inc., River Edge, NJ, 10.1142/9789812816689, URL http://dx.doi.org/10.1142/9789812816689, the theory of partial symmetries Martin [1994] Martin P (1994) Temperley-Lieb algebras for nonplanar statistical mechanics—the partition algebra construction. J Knot Theory Ramifications 3(1):51–82. 10.1142/S0218216594000071, URL http://dx.doi.org/10.1142/S0218216594000071 Martin [1996] Martin P (1996) The structure of the partition algebras. J Algebra 183(2):319–358. 10.1006/jabr.1996.0223, URL http://dx.doi.org/10.1006/jabr.1996.0223 Martin and Mazorchuk [2014] Martin P, Mazorchuk V (2014) On the representation theory of partial Brauer algebras. Q J Math 65(1):225–247. 10.1093/qmath/has043, URL https://doi.org/10.1093/qmath/has043 Martin [1990] Martin PP (1990) Representations of graph Temperley-Lieb algebras. Publ Res Inst Math Sci 26(3):485–503. 10.2977/prims/1195170958, URL https://doi.org/10.2977/prims/1195170958 Martin [2008] Martin S (2008) Schur algebras and representation theory, Cambridge Tracts in Mathematics, vol 112. Cambridge University Press, Cambridge, reprint of the 1993 original Munn [1955] Munn WD (1955) On semigroup algebras. Proc Cambridge Philos Soc 51:1–15 Munn [1957a] Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Jones VFR (1994) The Potts model and the symmetric group. In: Subfactors (Kyuzeso, 1993). World Sci. Publ., River Edge, NJ, p 259–267 Kudryavtseva and Mazorchuk [2008] Kudryavtseva G, Mazorchuk V (2008) Schur-Weyl dualities for symmetric inverse semigroups. J Pure Appl Algebra 212(8):1987–1995. 10.1016/j.jpaa.2007.12.004, URL https://doi.org/10.1016/j.jpaa.2007.12.004 Lawson [1998] Lawson MV (1998) Inverse semigroups. World Scientific Publishing Co. Inc., River Edge, NJ, 10.1142/9789812816689, URL http://dx.doi.org/10.1142/9789812816689, the theory of partial symmetries Martin [1994] Martin P (1994) Temperley-Lieb algebras for nonplanar statistical mechanics—the partition algebra construction. J Knot Theory Ramifications 3(1):51–82. 10.1142/S0218216594000071, URL http://dx.doi.org/10.1142/S0218216594000071 Martin [1996] Martin P (1996) The structure of the partition algebras. J Algebra 183(2):319–358. 10.1006/jabr.1996.0223, URL http://dx.doi.org/10.1006/jabr.1996.0223 Martin and Mazorchuk [2014] Martin P, Mazorchuk V (2014) On the representation theory of partial Brauer algebras. Q J Math 65(1):225–247. 10.1093/qmath/has043, URL https://doi.org/10.1093/qmath/has043 Martin [1990] Martin PP (1990) Representations of graph Temperley-Lieb algebras. Publ Res Inst Math Sci 26(3):485–503. 10.2977/prims/1195170958, URL https://doi.org/10.2977/prims/1195170958 Martin [2008] Martin S (2008) Schur algebras and representation theory, Cambridge Tracts in Mathematics, vol 112. Cambridge University Press, Cambridge, reprint of the 1993 original Munn [1955] Munn WD (1955) On semigroup algebras. Proc Cambridge Philos Soc 51:1–15 Munn [1957a] Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Kudryavtseva G, Mazorchuk V (2008) Schur-Weyl dualities for symmetric inverse semigroups. J Pure Appl Algebra 212(8):1987–1995. 10.1016/j.jpaa.2007.12.004, URL https://doi.org/10.1016/j.jpaa.2007.12.004 Lawson [1998] Lawson MV (1998) Inverse semigroups. World Scientific Publishing Co. Inc., River Edge, NJ, 10.1142/9789812816689, URL http://dx.doi.org/10.1142/9789812816689, the theory of partial symmetries Martin [1994] Martin P (1994) Temperley-Lieb algebras for nonplanar statistical mechanics—the partition algebra construction. J Knot Theory Ramifications 3(1):51–82. 10.1142/S0218216594000071, URL http://dx.doi.org/10.1142/S0218216594000071 Martin [1996] Martin P (1996) The structure of the partition algebras. J Algebra 183(2):319–358. 10.1006/jabr.1996.0223, URL http://dx.doi.org/10.1006/jabr.1996.0223 Martin and Mazorchuk [2014] Martin P, Mazorchuk V (2014) On the representation theory of partial Brauer algebras. Q J Math 65(1):225–247. 10.1093/qmath/has043, URL https://doi.org/10.1093/qmath/has043 Martin [1990] Martin PP (1990) Representations of graph Temperley-Lieb algebras. Publ Res Inst Math Sci 26(3):485–503. 10.2977/prims/1195170958, URL https://doi.org/10.2977/prims/1195170958 Martin [2008] Martin S (2008) Schur algebras and representation theory, Cambridge Tracts in Mathematics, vol 112. Cambridge University Press, Cambridge, reprint of the 1993 original Munn [1955] Munn WD (1955) On semigroup algebras. Proc Cambridge Philos Soc 51:1–15 Munn [1957a] Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Lawson MV (1998) Inverse semigroups. World Scientific Publishing Co. Inc., River Edge, NJ, 10.1142/9789812816689, URL http://dx.doi.org/10.1142/9789812816689, the theory of partial symmetries Martin [1994] Martin P (1994) Temperley-Lieb algebras for nonplanar statistical mechanics—the partition algebra construction. J Knot Theory Ramifications 3(1):51–82. 10.1142/S0218216594000071, URL http://dx.doi.org/10.1142/S0218216594000071 Martin [1996] Martin P (1996) The structure of the partition algebras. J Algebra 183(2):319–358. 10.1006/jabr.1996.0223, URL http://dx.doi.org/10.1006/jabr.1996.0223 Martin and Mazorchuk [2014] Martin P, Mazorchuk V (2014) On the representation theory of partial Brauer algebras. Q J Math 65(1):225–247. 10.1093/qmath/has043, URL https://doi.org/10.1093/qmath/has043 Martin [1990] Martin PP (1990) Representations of graph Temperley-Lieb algebras. Publ Res Inst Math Sci 26(3):485–503. 10.2977/prims/1195170958, URL https://doi.org/10.2977/prims/1195170958 Martin [2008] Martin S (2008) Schur algebras and representation theory, Cambridge Tracts in Mathematics, vol 112. Cambridge University Press, Cambridge, reprint of the 1993 original Munn [1955] Munn WD (1955) On semigroup algebras. Proc Cambridge Philos Soc 51:1–15 Munn [1957a] Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Martin P (1994) Temperley-Lieb algebras for nonplanar statistical mechanics—the partition algebra construction. J Knot Theory Ramifications 3(1):51–82. 10.1142/S0218216594000071, URL http://dx.doi.org/10.1142/S0218216594000071 Martin [1996] Martin P (1996) The structure of the partition algebras. J Algebra 183(2):319–358. 10.1006/jabr.1996.0223, URL http://dx.doi.org/10.1006/jabr.1996.0223 Martin and Mazorchuk [2014] Martin P, Mazorchuk V (2014) On the representation theory of partial Brauer algebras. Q J Math 65(1):225–247. 10.1093/qmath/has043, URL https://doi.org/10.1093/qmath/has043 Martin [1990] Martin PP (1990) Representations of graph Temperley-Lieb algebras. Publ Res Inst Math Sci 26(3):485–503. 10.2977/prims/1195170958, URL https://doi.org/10.2977/prims/1195170958 Martin [2008] Martin S (2008) Schur algebras and representation theory, Cambridge Tracts in Mathematics, vol 112. Cambridge University Press, Cambridge, reprint of the 1993 original Munn [1955] Munn WD (1955) On semigroup algebras. Proc Cambridge Philos Soc 51:1–15 Munn [1957a] Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Martin P (1996) The structure of the partition algebras. J Algebra 183(2):319–358. 10.1006/jabr.1996.0223, URL http://dx.doi.org/10.1006/jabr.1996.0223 Martin and Mazorchuk [2014] Martin P, Mazorchuk V (2014) On the representation theory of partial Brauer algebras. Q J Math 65(1):225–247. 10.1093/qmath/has043, URL https://doi.org/10.1093/qmath/has043 Martin [1990] Martin PP (1990) Representations of graph Temperley-Lieb algebras. Publ Res Inst Math Sci 26(3):485–503. 10.2977/prims/1195170958, URL https://doi.org/10.2977/prims/1195170958 Martin [2008] Martin S (2008) Schur algebras and representation theory, Cambridge Tracts in Mathematics, vol 112. Cambridge University Press, Cambridge, reprint of the 1993 original Munn [1955] Munn WD (1955) On semigroup algebras. Proc Cambridge Philos Soc 51:1–15 Munn [1957a] Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Martin P, Mazorchuk V (2014) On the representation theory of partial Brauer algebras. Q J Math 65(1):225–247. 10.1093/qmath/has043, URL https://doi.org/10.1093/qmath/has043 Martin [1990] Martin PP (1990) Representations of graph Temperley-Lieb algebras. Publ Res Inst Math Sci 26(3):485–503. 10.2977/prims/1195170958, URL https://doi.org/10.2977/prims/1195170958 Martin [2008] Martin S (2008) Schur algebras and representation theory, Cambridge Tracts in Mathematics, vol 112. Cambridge University Press, Cambridge, reprint of the 1993 original Munn [1955] Munn WD (1955) On semigroup algebras. Proc Cambridge Philos Soc 51:1–15 Munn [1957a] Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Martin PP (1990) Representations of graph Temperley-Lieb algebras. Publ Res Inst Math Sci 26(3):485–503. 10.2977/prims/1195170958, URL https://doi.org/10.2977/prims/1195170958 Martin [2008] Martin S (2008) Schur algebras and representation theory, Cambridge Tracts in Mathematics, vol 112. Cambridge University Press, Cambridge, reprint of the 1993 original Munn [1955] Munn WD (1955) On semigroup algebras. Proc Cambridge Philos Soc 51:1–15 Munn [1957a] Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Martin S (2008) Schur algebras and representation theory, Cambridge Tracts in Mathematics, vol 112. Cambridge University Press, Cambridge, reprint of the 1993 original Munn [1955] Munn WD (1955) On semigroup algebras. Proc Cambridge Philos Soc 51:1–15 Munn [1957a] Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Munn WD (1955) On semigroup algebras. Proc Cambridge Philos Soc 51:1–15 Munn [1957a] Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. 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J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. de Concini C, Procesi C (1976) A characteristic free approach to invariant theory. Advances in Math 21(3):330–354 Curtis and Reiner [1981] Curtis CW, Reiner I (1981) Methods of representation theory. Vol. I. John Wiley & Sons Inc., New York, with applications to finite groups and orders, Pure and Applied Mathematics, A Wiley-Interscience Publication Dipper et al [2008] Dipper R, Doty S, Hu J (2008) Brauer algebras, symplectic Schur algebras and Schur-Weyl duality. Trans Amer Math Soc 360(1):189–213. 10.1090/S0002-9947-07-04179-7, URL https://doi.org/10.1090/S0002-9947-07-04179-7 Doty and Hu [2009] Doty S, Hu J (2009) Schur-Weyl duality for orthogonal groups. Proc Lond Math Soc (3) 98(3):679–713. 10.1112/plms/pdn044, URL https://doi.org/10.1112/plms/pdn044 Goodman and Wallach [2009] Goodman R, Wallach NR (2009) Symmetry, representations, and invariants, Graduate Texts in Mathematics, vol 255. Springer, Dordrecht, 10.1007/978-0-387-79852-3, URL https://doi.org/10.1007/978-0-387-79852-3 Graham and Lehrer [1996] Graham JJ, Lehrer GI (1996) Cellular algebras. Invent Math 123(1):1–34. 10.1007/BF01232365, URL http://dx.doi.org/10.1007/BF01232365 Green [1980] Green JA (1980) Polynomial representations of GLnsubscriptGL𝑛{\rm GL}_{n}roman_GL start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT, Lecture Notes in Mathematics, vol 830. Springer-Verlag, Berlin Grood [2006] Grood C (2006) The rook partition algebra. J Combin Theory Ser A 113(2):325–351. 10.1016/j.jcta.2005.03.006, URL https://doi.org/10.1016/j.jcta.2005.03.006 Halverson [1996] Halverson T (1996) Characters of the centralizer algebras of mixed tensor representations of Gl(r,ℂ)Gl𝑟ℂ{\rm Gl}(r,\mathbb{C})roman_Gl ( italic_r , blackboard_C ) and the quantum group 𝔘q(ℊl(r,ℂ))subscript𝔘𝑞ℊ𝑙𝑟ℂ\mathfrak{U}_{q}({\mathcal{g}l}(r,\mathbb{C}))fraktur_U start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ( caligraphic_g italic_l ( italic_r , blackboard_C ) ). Pacific J Math 174(2):359–410. URL http://projecteuclid.org/euclid.pjm/1102365176 Halverson [2004] Halverson T (2004) Representations of the q𝑞qitalic_q-rook monoid. J Algebra 273(1):227–251. 10.1016/j.jalgebra.2003.11.002, URL http://dx.doi.org/10.1016/j.jalgebra.2003.11.002 Halverson and delMas [2014] Halverson T, delMas E (2014) Representations of the Rook-Brauer algebra. Comm Algebra 42(1):423–443. 10.1080/00927872.2012.716120, URL https://doi.org/10.1080/00927872.2012.716120 Halverson and Ram [2005] Halverson T, Ram A (2005) Partition algebras. European J Combin 26(6):869–921. 10.1016/j.ejc.2004.06.005, URL http://dx.doi.org/10.1016/j.ejc.2004.06.005 Jacobson [1989] Jacobson N (1989) Basic algebra. II, 2nd edn. W. H. Freeman and Company, New York Jones [1994] Jones VFR (1994) The Potts model and the symmetric group. In: Subfactors (Kyuzeso, 1993). World Sci. Publ., River Edge, NJ, p 259–267 Kudryavtseva and Mazorchuk [2008] Kudryavtseva G, Mazorchuk V (2008) Schur-Weyl dualities for symmetric inverse semigroups. J Pure Appl Algebra 212(8):1987–1995. 10.1016/j.jpaa.2007.12.004, URL https://doi.org/10.1016/j.jpaa.2007.12.004 Lawson [1998] Lawson MV (1998) Inverse semigroups. World Scientific Publishing Co. Inc., River Edge, NJ, 10.1142/9789812816689, URL http://dx.doi.org/10.1142/9789812816689, the theory of partial symmetries Martin [1994] Martin P (1994) Temperley-Lieb algebras for nonplanar statistical mechanics—the partition algebra construction. J Knot Theory Ramifications 3(1):51–82. 10.1142/S0218216594000071, URL http://dx.doi.org/10.1142/S0218216594000071 Martin [1996] Martin P (1996) The structure of the partition algebras. J Algebra 183(2):319–358. 10.1006/jabr.1996.0223, URL http://dx.doi.org/10.1006/jabr.1996.0223 Martin and Mazorchuk [2014] Martin P, Mazorchuk V (2014) On the representation theory of partial Brauer algebras. Q J Math 65(1):225–247. 10.1093/qmath/has043, URL https://doi.org/10.1093/qmath/has043 Martin [1990] Martin PP (1990) Representations of graph Temperley-Lieb algebras. Publ Res Inst Math Sci 26(3):485–503. 10.2977/prims/1195170958, URL https://doi.org/10.2977/prims/1195170958 Martin [2008] Martin S (2008) Schur algebras and representation theory, Cambridge Tracts in Mathematics, vol 112. Cambridge University Press, Cambridge, reprint of the 1993 original Munn [1955] Munn WD (1955) On semigroup algebras. Proc Cambridge Philos Soc 51:1–15 Munn [1957a] Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Curtis CW, Reiner I (1981) Methods of representation theory. Vol. I. John Wiley & Sons Inc., New York, with applications to finite groups and orders, Pure and Applied Mathematics, A Wiley-Interscience Publication Dipper et al [2008] Dipper R, Doty S, Hu J (2008) Brauer algebras, symplectic Schur algebras and Schur-Weyl duality. Trans Amer Math Soc 360(1):189–213. 10.1090/S0002-9947-07-04179-7, URL https://doi.org/10.1090/S0002-9947-07-04179-7 Doty and Hu [2009] Doty S, Hu J (2009) Schur-Weyl duality for orthogonal groups. Proc Lond Math Soc (3) 98(3):679–713. 10.1112/plms/pdn044, URL https://doi.org/10.1112/plms/pdn044 Goodman and Wallach [2009] Goodman R, Wallach NR (2009) Symmetry, representations, and invariants, Graduate Texts in Mathematics, vol 255. Springer, Dordrecht, 10.1007/978-0-387-79852-3, URL https://doi.org/10.1007/978-0-387-79852-3 Graham and Lehrer [1996] Graham JJ, Lehrer GI (1996) Cellular algebras. Invent Math 123(1):1–34. 10.1007/BF01232365, URL http://dx.doi.org/10.1007/BF01232365 Green [1980] Green JA (1980) Polynomial representations of GLnsubscriptGL𝑛{\rm GL}_{n}roman_GL start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT, Lecture Notes in Mathematics, vol 830. Springer-Verlag, Berlin Grood [2006] Grood C (2006) The rook partition algebra. J Combin Theory Ser A 113(2):325–351. 10.1016/j.jcta.2005.03.006, URL https://doi.org/10.1016/j.jcta.2005.03.006 Halverson [1996] Halverson T (1996) Characters of the centralizer algebras of mixed tensor representations of Gl(r,ℂ)Gl𝑟ℂ{\rm Gl}(r,\mathbb{C})roman_Gl ( italic_r , blackboard_C ) and the quantum group 𝔘q(ℊl(r,ℂ))subscript𝔘𝑞ℊ𝑙𝑟ℂ\mathfrak{U}_{q}({\mathcal{g}l}(r,\mathbb{C}))fraktur_U start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ( caligraphic_g italic_l ( italic_r , blackboard_C ) ). Pacific J Math 174(2):359–410. URL http://projecteuclid.org/euclid.pjm/1102365176 Halverson [2004] Halverson T (2004) Representations of the q𝑞qitalic_q-rook monoid. J Algebra 273(1):227–251. 10.1016/j.jalgebra.2003.11.002, URL http://dx.doi.org/10.1016/j.jalgebra.2003.11.002 Halverson and delMas [2014] Halverson T, delMas E (2014) Representations of the Rook-Brauer algebra. Comm Algebra 42(1):423–443. 10.1080/00927872.2012.716120, URL https://doi.org/10.1080/00927872.2012.716120 Halverson and Ram [2005] Halverson T, Ram A (2005) Partition algebras. European J Combin 26(6):869–921. 10.1016/j.ejc.2004.06.005, URL http://dx.doi.org/10.1016/j.ejc.2004.06.005 Jacobson [1989] Jacobson N (1989) Basic algebra. II, 2nd edn. W. H. Freeman and Company, New York Jones [1994] Jones VFR (1994) The Potts model and the symmetric group. In: Subfactors (Kyuzeso, 1993). World Sci. Publ., River Edge, NJ, p 259–267 Kudryavtseva and Mazorchuk [2008] Kudryavtseva G, Mazorchuk V (2008) Schur-Weyl dualities for symmetric inverse semigroups. J Pure Appl Algebra 212(8):1987–1995. 10.1016/j.jpaa.2007.12.004, URL https://doi.org/10.1016/j.jpaa.2007.12.004 Lawson [1998] Lawson MV (1998) Inverse semigroups. World Scientific Publishing Co. Inc., River Edge, NJ, 10.1142/9789812816689, URL http://dx.doi.org/10.1142/9789812816689, the theory of partial symmetries Martin [1994] Martin P (1994) Temperley-Lieb algebras for nonplanar statistical mechanics—the partition algebra construction. J Knot Theory Ramifications 3(1):51–82. 10.1142/S0218216594000071, URL http://dx.doi.org/10.1142/S0218216594000071 Martin [1996] Martin P (1996) The structure of the partition algebras. J Algebra 183(2):319–358. 10.1006/jabr.1996.0223, URL http://dx.doi.org/10.1006/jabr.1996.0223 Martin and Mazorchuk [2014] Martin P, Mazorchuk V (2014) On the representation theory of partial Brauer algebras. Q J Math 65(1):225–247. 10.1093/qmath/has043, URL https://doi.org/10.1093/qmath/has043 Martin [1990] Martin PP (1990) Representations of graph Temperley-Lieb algebras. Publ Res Inst Math Sci 26(3):485–503. 10.2977/prims/1195170958, URL https://doi.org/10.2977/prims/1195170958 Martin [2008] Martin S (2008) Schur algebras and representation theory, Cambridge Tracts in Mathematics, vol 112. Cambridge University Press, Cambridge, reprint of the 1993 original Munn [1955] Munn WD (1955) On semigroup algebras. Proc Cambridge Philos Soc 51:1–15 Munn [1957a] Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Dipper R, Doty S, Hu J (2008) Brauer algebras, symplectic Schur algebras and Schur-Weyl duality. Trans Amer Math Soc 360(1):189–213. 10.1090/S0002-9947-07-04179-7, URL https://doi.org/10.1090/S0002-9947-07-04179-7 Doty and Hu [2009] Doty S, Hu J (2009) Schur-Weyl duality for orthogonal groups. Proc Lond Math Soc (3) 98(3):679–713. 10.1112/plms/pdn044, URL https://doi.org/10.1112/plms/pdn044 Goodman and Wallach [2009] Goodman R, Wallach NR (2009) Symmetry, representations, and invariants, Graduate Texts in Mathematics, vol 255. Springer, Dordrecht, 10.1007/978-0-387-79852-3, URL https://doi.org/10.1007/978-0-387-79852-3 Graham and Lehrer [1996] Graham JJ, Lehrer GI (1996) Cellular algebras. Invent Math 123(1):1–34. 10.1007/BF01232365, URL http://dx.doi.org/10.1007/BF01232365 Green [1980] Green JA (1980) Polynomial representations of GLnsubscriptGL𝑛{\rm GL}_{n}roman_GL start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT, Lecture Notes in Mathematics, vol 830. Springer-Verlag, Berlin Grood [2006] Grood C (2006) The rook partition algebra. J Combin Theory Ser A 113(2):325–351. 10.1016/j.jcta.2005.03.006, URL https://doi.org/10.1016/j.jcta.2005.03.006 Halverson [1996] Halverson T (1996) Characters of the centralizer algebras of mixed tensor representations of Gl(r,ℂ)Gl𝑟ℂ{\rm Gl}(r,\mathbb{C})roman_Gl ( italic_r , blackboard_C ) and the quantum group 𝔘q(ℊl(r,ℂ))subscript𝔘𝑞ℊ𝑙𝑟ℂ\mathfrak{U}_{q}({\mathcal{g}l}(r,\mathbb{C}))fraktur_U start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ( caligraphic_g italic_l ( italic_r , blackboard_C ) ). Pacific J Math 174(2):359–410. URL http://projecteuclid.org/euclid.pjm/1102365176 Halverson [2004] Halverson T (2004) Representations of the q𝑞qitalic_q-rook monoid. J Algebra 273(1):227–251. 10.1016/j.jalgebra.2003.11.002, URL http://dx.doi.org/10.1016/j.jalgebra.2003.11.002 Halverson and delMas [2014] Halverson T, delMas E (2014) Representations of the Rook-Brauer algebra. Comm Algebra 42(1):423–443. 10.1080/00927872.2012.716120, URL https://doi.org/10.1080/00927872.2012.716120 Halverson and Ram [2005] Halverson T, Ram A (2005) Partition algebras. European J Combin 26(6):869–921. 10.1016/j.ejc.2004.06.005, URL http://dx.doi.org/10.1016/j.ejc.2004.06.005 Jacobson [1989] Jacobson N (1989) Basic algebra. II, 2nd edn. W. H. Freeman and Company, New York Jones [1994] Jones VFR (1994) The Potts model and the symmetric group. In: Subfactors (Kyuzeso, 1993). World Sci. Publ., River Edge, NJ, p 259–267 Kudryavtseva and Mazorchuk [2008] Kudryavtseva G, Mazorchuk V (2008) Schur-Weyl dualities for symmetric inverse semigroups. J Pure Appl Algebra 212(8):1987–1995. 10.1016/j.jpaa.2007.12.004, URL https://doi.org/10.1016/j.jpaa.2007.12.004 Lawson [1998] Lawson MV (1998) Inverse semigroups. World Scientific Publishing Co. Inc., River Edge, NJ, 10.1142/9789812816689, URL http://dx.doi.org/10.1142/9789812816689, the theory of partial symmetries Martin [1994] Martin P (1994) Temperley-Lieb algebras for nonplanar statistical mechanics—the partition algebra construction. J Knot Theory Ramifications 3(1):51–82. 10.1142/S0218216594000071, URL http://dx.doi.org/10.1142/S0218216594000071 Martin [1996] Martin P (1996) The structure of the partition algebras. J Algebra 183(2):319–358. 10.1006/jabr.1996.0223, URL http://dx.doi.org/10.1006/jabr.1996.0223 Martin and Mazorchuk [2014] Martin P, Mazorchuk V (2014) On the representation theory of partial Brauer algebras. Q J Math 65(1):225–247. 10.1093/qmath/has043, URL https://doi.org/10.1093/qmath/has043 Martin [1990] Martin PP (1990) Representations of graph Temperley-Lieb algebras. Publ Res Inst Math Sci 26(3):485–503. 10.2977/prims/1195170958, URL https://doi.org/10.2977/prims/1195170958 Martin [2008] Martin S (2008) Schur algebras and representation theory, Cambridge Tracts in Mathematics, vol 112. Cambridge University Press, Cambridge, reprint of the 1993 original Munn [1955] Munn WD (1955) On semigroup algebras. Proc Cambridge Philos Soc 51:1–15 Munn [1957a] Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Doty S, Hu J (2009) Schur-Weyl duality for orthogonal groups. Proc Lond Math Soc (3) 98(3):679–713. 10.1112/plms/pdn044, URL https://doi.org/10.1112/plms/pdn044 Goodman and Wallach [2009] Goodman R, Wallach NR (2009) Symmetry, representations, and invariants, Graduate Texts in Mathematics, vol 255. Springer, Dordrecht, 10.1007/978-0-387-79852-3, URL https://doi.org/10.1007/978-0-387-79852-3 Graham and Lehrer [1996] Graham JJ, Lehrer GI (1996) Cellular algebras. Invent Math 123(1):1–34. 10.1007/BF01232365, URL http://dx.doi.org/10.1007/BF01232365 Green [1980] Green JA (1980) Polynomial representations of GLnsubscriptGL𝑛{\rm GL}_{n}roman_GL start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT, Lecture Notes in Mathematics, vol 830. Springer-Verlag, Berlin Grood [2006] Grood C (2006) The rook partition algebra. J Combin Theory Ser A 113(2):325–351. 10.1016/j.jcta.2005.03.006, URL https://doi.org/10.1016/j.jcta.2005.03.006 Halverson [1996] Halverson T (1996) Characters of the centralizer algebras of mixed tensor representations of Gl(r,ℂ)Gl𝑟ℂ{\rm Gl}(r,\mathbb{C})roman_Gl ( italic_r , blackboard_C ) and the quantum group 𝔘q(ℊl(r,ℂ))subscript𝔘𝑞ℊ𝑙𝑟ℂ\mathfrak{U}_{q}({\mathcal{g}l}(r,\mathbb{C}))fraktur_U start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ( caligraphic_g italic_l ( italic_r , blackboard_C ) ). Pacific J Math 174(2):359–410. URL http://projecteuclid.org/euclid.pjm/1102365176 Halverson [2004] Halverson T (2004) Representations of the q𝑞qitalic_q-rook monoid. J Algebra 273(1):227–251. 10.1016/j.jalgebra.2003.11.002, URL http://dx.doi.org/10.1016/j.jalgebra.2003.11.002 Halverson and delMas [2014] Halverson T, delMas E (2014) Representations of the Rook-Brauer algebra. Comm Algebra 42(1):423–443. 10.1080/00927872.2012.716120, URL https://doi.org/10.1080/00927872.2012.716120 Halverson and Ram [2005] Halverson T, Ram A (2005) Partition algebras. European J Combin 26(6):869–921. 10.1016/j.ejc.2004.06.005, URL http://dx.doi.org/10.1016/j.ejc.2004.06.005 Jacobson [1989] Jacobson N (1989) Basic algebra. II, 2nd edn. W. H. Freeman and Company, New York Jones [1994] Jones VFR (1994) The Potts model and the symmetric group. In: Subfactors (Kyuzeso, 1993). World Sci. Publ., River Edge, NJ, p 259–267 Kudryavtseva and Mazorchuk [2008] Kudryavtseva G, Mazorchuk V (2008) Schur-Weyl dualities for symmetric inverse semigroups. J Pure Appl Algebra 212(8):1987–1995. 10.1016/j.jpaa.2007.12.004, URL https://doi.org/10.1016/j.jpaa.2007.12.004 Lawson [1998] Lawson MV (1998) Inverse semigroups. World Scientific Publishing Co. Inc., River Edge, NJ, 10.1142/9789812816689, URL http://dx.doi.org/10.1142/9789812816689, the theory of partial symmetries Martin [1994] Martin P (1994) Temperley-Lieb algebras for nonplanar statistical mechanics—the partition algebra construction. J Knot Theory Ramifications 3(1):51–82. 10.1142/S0218216594000071, URL http://dx.doi.org/10.1142/S0218216594000071 Martin [1996] Martin P (1996) The structure of the partition algebras. J Algebra 183(2):319–358. 10.1006/jabr.1996.0223, URL http://dx.doi.org/10.1006/jabr.1996.0223 Martin and Mazorchuk [2014] Martin P, Mazorchuk V (2014) On the representation theory of partial Brauer algebras. Q J Math 65(1):225–247. 10.1093/qmath/has043, URL https://doi.org/10.1093/qmath/has043 Martin [1990] Martin PP (1990) Representations of graph Temperley-Lieb algebras. Publ Res Inst Math Sci 26(3):485–503. 10.2977/prims/1195170958, URL https://doi.org/10.2977/prims/1195170958 Martin [2008] Martin S (2008) Schur algebras and representation theory, Cambridge Tracts in Mathematics, vol 112. Cambridge University Press, Cambridge, reprint of the 1993 original Munn [1955] Munn WD (1955) On semigroup algebras. Proc Cambridge Philos Soc 51:1–15 Munn [1957a] Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Goodman R, Wallach NR (2009) Symmetry, representations, and invariants, Graduate Texts in Mathematics, vol 255. Springer, Dordrecht, 10.1007/978-0-387-79852-3, URL https://doi.org/10.1007/978-0-387-79852-3 Graham and Lehrer [1996] Graham JJ, Lehrer GI (1996) Cellular algebras. Invent Math 123(1):1–34. 10.1007/BF01232365, URL http://dx.doi.org/10.1007/BF01232365 Green [1980] Green JA (1980) Polynomial representations of GLnsubscriptGL𝑛{\rm GL}_{n}roman_GL start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT, Lecture Notes in Mathematics, vol 830. Springer-Verlag, Berlin Grood [2006] Grood C (2006) The rook partition algebra. J Combin Theory Ser A 113(2):325–351. 10.1016/j.jcta.2005.03.006, URL https://doi.org/10.1016/j.jcta.2005.03.006 Halverson [1996] Halverson T (1996) Characters of the centralizer algebras of mixed tensor representations of Gl(r,ℂ)Gl𝑟ℂ{\rm Gl}(r,\mathbb{C})roman_Gl ( italic_r , blackboard_C ) and the quantum group 𝔘q(ℊl(r,ℂ))subscript𝔘𝑞ℊ𝑙𝑟ℂ\mathfrak{U}_{q}({\mathcal{g}l}(r,\mathbb{C}))fraktur_U start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ( caligraphic_g italic_l ( italic_r , blackboard_C ) ). Pacific J Math 174(2):359–410. URL http://projecteuclid.org/euclid.pjm/1102365176 Halverson [2004] Halverson T (2004) Representations of the q𝑞qitalic_q-rook monoid. J Algebra 273(1):227–251. 10.1016/j.jalgebra.2003.11.002, URL http://dx.doi.org/10.1016/j.jalgebra.2003.11.002 Halverson and delMas [2014] Halverson T, delMas E (2014) Representations of the Rook-Brauer algebra. Comm Algebra 42(1):423–443. 10.1080/00927872.2012.716120, URL https://doi.org/10.1080/00927872.2012.716120 Halverson and Ram [2005] Halverson T, Ram A (2005) Partition algebras. European J Combin 26(6):869–921. 10.1016/j.ejc.2004.06.005, URL http://dx.doi.org/10.1016/j.ejc.2004.06.005 Jacobson [1989] Jacobson N (1989) Basic algebra. II, 2nd edn. W. H. Freeman and Company, New York Jones [1994] Jones VFR (1994) The Potts model and the symmetric group. In: Subfactors (Kyuzeso, 1993). World Sci. Publ., River Edge, NJ, p 259–267 Kudryavtseva and Mazorchuk [2008] Kudryavtseva G, Mazorchuk V (2008) Schur-Weyl dualities for symmetric inverse semigroups. J Pure Appl Algebra 212(8):1987–1995. 10.1016/j.jpaa.2007.12.004, URL https://doi.org/10.1016/j.jpaa.2007.12.004 Lawson [1998] Lawson MV (1998) Inverse semigroups. World Scientific Publishing Co. Inc., River Edge, NJ, 10.1142/9789812816689, URL http://dx.doi.org/10.1142/9789812816689, the theory of partial symmetries Martin [1994] Martin P (1994) Temperley-Lieb algebras for nonplanar statistical mechanics—the partition algebra construction. J Knot Theory Ramifications 3(1):51–82. 10.1142/S0218216594000071, URL http://dx.doi.org/10.1142/S0218216594000071 Martin [1996] Martin P (1996) The structure of the partition algebras. J Algebra 183(2):319–358. 10.1006/jabr.1996.0223, URL http://dx.doi.org/10.1006/jabr.1996.0223 Martin and Mazorchuk [2014] Martin P, Mazorchuk V (2014) On the representation theory of partial Brauer algebras. Q J Math 65(1):225–247. 10.1093/qmath/has043, URL https://doi.org/10.1093/qmath/has043 Martin [1990] Martin PP (1990) Representations of graph Temperley-Lieb algebras. Publ Res Inst Math Sci 26(3):485–503. 10.2977/prims/1195170958, URL https://doi.org/10.2977/prims/1195170958 Martin [2008] Martin S (2008) Schur algebras and representation theory, Cambridge Tracts in Mathematics, vol 112. Cambridge University Press, Cambridge, reprint of the 1993 original Munn [1955] Munn WD (1955) On semigroup algebras. Proc Cambridge Philos Soc 51:1–15 Munn [1957a] Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Graham JJ, Lehrer GI (1996) Cellular algebras. Invent Math 123(1):1–34. 10.1007/BF01232365, URL http://dx.doi.org/10.1007/BF01232365 Green [1980] Green JA (1980) Polynomial representations of GLnsubscriptGL𝑛{\rm GL}_{n}roman_GL start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT, Lecture Notes in Mathematics, vol 830. Springer-Verlag, Berlin Grood [2006] Grood C (2006) The rook partition algebra. J Combin Theory Ser A 113(2):325–351. 10.1016/j.jcta.2005.03.006, URL https://doi.org/10.1016/j.jcta.2005.03.006 Halverson [1996] Halverson T (1996) Characters of the centralizer algebras of mixed tensor representations of Gl(r,ℂ)Gl𝑟ℂ{\rm Gl}(r,\mathbb{C})roman_Gl ( italic_r , blackboard_C ) and the quantum group 𝔘q(ℊl(r,ℂ))subscript𝔘𝑞ℊ𝑙𝑟ℂ\mathfrak{U}_{q}({\mathcal{g}l}(r,\mathbb{C}))fraktur_U start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ( caligraphic_g italic_l ( italic_r , blackboard_C ) ). Pacific J Math 174(2):359–410. URL http://projecteuclid.org/euclid.pjm/1102365176 Halverson [2004] Halverson T (2004) Representations of the q𝑞qitalic_q-rook monoid. J Algebra 273(1):227–251. 10.1016/j.jalgebra.2003.11.002, URL http://dx.doi.org/10.1016/j.jalgebra.2003.11.002 Halverson and delMas [2014] Halverson T, delMas E (2014) Representations of the Rook-Brauer algebra. Comm Algebra 42(1):423–443. 10.1080/00927872.2012.716120, URL https://doi.org/10.1080/00927872.2012.716120 Halverson and Ram [2005] Halverson T, Ram A (2005) Partition algebras. European J Combin 26(6):869–921. 10.1016/j.ejc.2004.06.005, URL http://dx.doi.org/10.1016/j.ejc.2004.06.005 Jacobson [1989] Jacobson N (1989) Basic algebra. II, 2nd edn. W. H. Freeman and Company, New York Jones [1994] Jones VFR (1994) The Potts model and the symmetric group. In: Subfactors (Kyuzeso, 1993). World Sci. Publ., River Edge, NJ, p 259–267 Kudryavtseva and Mazorchuk [2008] Kudryavtseva G, Mazorchuk V (2008) Schur-Weyl dualities for symmetric inverse semigroups. J Pure Appl Algebra 212(8):1987–1995. 10.1016/j.jpaa.2007.12.004, URL https://doi.org/10.1016/j.jpaa.2007.12.004 Lawson [1998] Lawson MV (1998) Inverse semigroups. World Scientific Publishing Co. Inc., River Edge, NJ, 10.1142/9789812816689, URL http://dx.doi.org/10.1142/9789812816689, the theory of partial symmetries Martin [1994] Martin P (1994) Temperley-Lieb algebras for nonplanar statistical mechanics—the partition algebra construction. J Knot Theory Ramifications 3(1):51–82. 10.1142/S0218216594000071, URL http://dx.doi.org/10.1142/S0218216594000071 Martin [1996] Martin P (1996) The structure of the partition algebras. J Algebra 183(2):319–358. 10.1006/jabr.1996.0223, URL http://dx.doi.org/10.1006/jabr.1996.0223 Martin and Mazorchuk [2014] Martin P, Mazorchuk V (2014) On the representation theory of partial Brauer algebras. Q J Math 65(1):225–247. 10.1093/qmath/has043, URL https://doi.org/10.1093/qmath/has043 Martin [1990] Martin PP (1990) Representations of graph Temperley-Lieb algebras. Publ Res Inst Math Sci 26(3):485–503. 10.2977/prims/1195170958, URL https://doi.org/10.2977/prims/1195170958 Martin [2008] Martin S (2008) Schur algebras and representation theory, Cambridge Tracts in Mathematics, vol 112. Cambridge University Press, Cambridge, reprint of the 1993 original Munn [1955] Munn WD (1955) On semigroup algebras. Proc Cambridge Philos Soc 51:1–15 Munn [1957a] Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Green JA (1980) Polynomial representations of GLnsubscriptGL𝑛{\rm GL}_{n}roman_GL start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT, Lecture Notes in Mathematics, vol 830. Springer-Verlag, Berlin Grood [2006] Grood C (2006) The rook partition algebra. J Combin Theory Ser A 113(2):325–351. 10.1016/j.jcta.2005.03.006, URL https://doi.org/10.1016/j.jcta.2005.03.006 Halverson [1996] Halverson T (1996) Characters of the centralizer algebras of mixed tensor representations of Gl(r,ℂ)Gl𝑟ℂ{\rm Gl}(r,\mathbb{C})roman_Gl ( italic_r , blackboard_C ) and the quantum group 𝔘q(ℊl(r,ℂ))subscript𝔘𝑞ℊ𝑙𝑟ℂ\mathfrak{U}_{q}({\mathcal{g}l}(r,\mathbb{C}))fraktur_U start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ( caligraphic_g italic_l ( italic_r , blackboard_C ) ). Pacific J Math 174(2):359–410. URL http://projecteuclid.org/euclid.pjm/1102365176 Halverson [2004] Halverson T (2004) Representations of the q𝑞qitalic_q-rook monoid. J Algebra 273(1):227–251. 10.1016/j.jalgebra.2003.11.002, URL http://dx.doi.org/10.1016/j.jalgebra.2003.11.002 Halverson and delMas [2014] Halverson T, delMas E (2014) Representations of the Rook-Brauer algebra. Comm Algebra 42(1):423–443. 10.1080/00927872.2012.716120, URL https://doi.org/10.1080/00927872.2012.716120 Halverson and Ram [2005] Halverson T, Ram A (2005) Partition algebras. European J Combin 26(6):869–921. 10.1016/j.ejc.2004.06.005, URL http://dx.doi.org/10.1016/j.ejc.2004.06.005 Jacobson [1989] Jacobson N (1989) Basic algebra. II, 2nd edn. W. H. Freeman and Company, New York Jones [1994] Jones VFR (1994) The Potts model and the symmetric group. In: Subfactors (Kyuzeso, 1993). World Sci. Publ., River Edge, NJ, p 259–267 Kudryavtseva and Mazorchuk [2008] Kudryavtseva G, Mazorchuk V (2008) Schur-Weyl dualities for symmetric inverse semigroups. J Pure Appl Algebra 212(8):1987–1995. 10.1016/j.jpaa.2007.12.004, URL https://doi.org/10.1016/j.jpaa.2007.12.004 Lawson [1998] Lawson MV (1998) Inverse semigroups. World Scientific Publishing Co. Inc., River Edge, NJ, 10.1142/9789812816689, URL http://dx.doi.org/10.1142/9789812816689, the theory of partial symmetries Martin [1994] Martin P (1994) Temperley-Lieb algebras for nonplanar statistical mechanics—the partition algebra construction. J Knot Theory Ramifications 3(1):51–82. 10.1142/S0218216594000071, URL http://dx.doi.org/10.1142/S0218216594000071 Martin [1996] Martin P (1996) The structure of the partition algebras. J Algebra 183(2):319–358. 10.1006/jabr.1996.0223, URL http://dx.doi.org/10.1006/jabr.1996.0223 Martin and Mazorchuk [2014] Martin P, Mazorchuk V (2014) On the representation theory of partial Brauer algebras. Q J Math 65(1):225–247. 10.1093/qmath/has043, URL https://doi.org/10.1093/qmath/has043 Martin [1990] Martin PP (1990) Representations of graph Temperley-Lieb algebras. Publ Res Inst Math Sci 26(3):485–503. 10.2977/prims/1195170958, URL https://doi.org/10.2977/prims/1195170958 Martin [2008] Martin S (2008) Schur algebras and representation theory, Cambridge Tracts in Mathematics, vol 112. Cambridge University Press, Cambridge, reprint of the 1993 original Munn [1955] Munn WD (1955) On semigroup algebras. Proc Cambridge Philos Soc 51:1–15 Munn [1957a] Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Grood C (2006) The rook partition algebra. J Combin Theory Ser A 113(2):325–351. 10.1016/j.jcta.2005.03.006, URL https://doi.org/10.1016/j.jcta.2005.03.006 Halverson [1996] Halverson T (1996) Characters of the centralizer algebras of mixed tensor representations of Gl(r,ℂ)Gl𝑟ℂ{\rm Gl}(r,\mathbb{C})roman_Gl ( italic_r , blackboard_C ) and the quantum group 𝔘q(ℊl(r,ℂ))subscript𝔘𝑞ℊ𝑙𝑟ℂ\mathfrak{U}_{q}({\mathcal{g}l}(r,\mathbb{C}))fraktur_U start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ( caligraphic_g italic_l ( italic_r , blackboard_C ) ). Pacific J Math 174(2):359–410. URL http://projecteuclid.org/euclid.pjm/1102365176 Halverson [2004] Halverson T (2004) Representations of the q𝑞qitalic_q-rook monoid. J Algebra 273(1):227–251. 10.1016/j.jalgebra.2003.11.002, URL http://dx.doi.org/10.1016/j.jalgebra.2003.11.002 Halverson and delMas [2014] Halverson T, delMas E (2014) Representations of the Rook-Brauer algebra. Comm Algebra 42(1):423–443. 10.1080/00927872.2012.716120, URL https://doi.org/10.1080/00927872.2012.716120 Halverson and Ram [2005] Halverson T, Ram A (2005) Partition algebras. European J Combin 26(6):869–921. 10.1016/j.ejc.2004.06.005, URL http://dx.doi.org/10.1016/j.ejc.2004.06.005 Jacobson [1989] Jacobson N (1989) Basic algebra. II, 2nd edn. W. H. Freeman and Company, New York Jones [1994] Jones VFR (1994) The Potts model and the symmetric group. In: Subfactors (Kyuzeso, 1993). World Sci. Publ., River Edge, NJ, p 259–267 Kudryavtseva and Mazorchuk [2008] Kudryavtseva G, Mazorchuk V (2008) Schur-Weyl dualities for symmetric inverse semigroups. J Pure Appl Algebra 212(8):1987–1995. 10.1016/j.jpaa.2007.12.004, URL https://doi.org/10.1016/j.jpaa.2007.12.004 Lawson [1998] Lawson MV (1998) Inverse semigroups. World Scientific Publishing Co. Inc., River Edge, NJ, 10.1142/9789812816689, URL http://dx.doi.org/10.1142/9789812816689, the theory of partial symmetries Martin [1994] Martin P (1994) Temperley-Lieb algebras for nonplanar statistical mechanics—the partition algebra construction. J Knot Theory Ramifications 3(1):51–82. 10.1142/S0218216594000071, URL http://dx.doi.org/10.1142/S0218216594000071 Martin [1996] Martin P (1996) The structure of the partition algebras. J Algebra 183(2):319–358. 10.1006/jabr.1996.0223, URL http://dx.doi.org/10.1006/jabr.1996.0223 Martin and Mazorchuk [2014] Martin P, Mazorchuk V (2014) On the representation theory of partial Brauer algebras. Q J Math 65(1):225–247. 10.1093/qmath/has043, URL https://doi.org/10.1093/qmath/has043 Martin [1990] Martin PP (1990) Representations of graph Temperley-Lieb algebras. Publ Res Inst Math Sci 26(3):485–503. 10.2977/prims/1195170958, URL https://doi.org/10.2977/prims/1195170958 Martin [2008] Martin S (2008) Schur algebras and representation theory, Cambridge Tracts in Mathematics, vol 112. Cambridge University Press, Cambridge, reprint of the 1993 original Munn [1955] Munn WD (1955) On semigroup algebras. Proc Cambridge Philos Soc 51:1–15 Munn [1957a] Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Halverson T (1996) Characters of the centralizer algebras of mixed tensor representations of Gl(r,ℂ)Gl𝑟ℂ{\rm Gl}(r,\mathbb{C})roman_Gl ( italic_r , blackboard_C ) and the quantum group 𝔘q(ℊl(r,ℂ))subscript𝔘𝑞ℊ𝑙𝑟ℂ\mathfrak{U}_{q}({\mathcal{g}l}(r,\mathbb{C}))fraktur_U start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ( caligraphic_g italic_l ( italic_r , blackboard_C ) ). Pacific J Math 174(2):359–410. URL http://projecteuclid.org/euclid.pjm/1102365176 Halverson [2004] Halverson T (2004) Representations of the q𝑞qitalic_q-rook monoid. J Algebra 273(1):227–251. 10.1016/j.jalgebra.2003.11.002, URL http://dx.doi.org/10.1016/j.jalgebra.2003.11.002 Halverson and delMas [2014] Halverson T, delMas E (2014) Representations of the Rook-Brauer algebra. Comm Algebra 42(1):423–443. 10.1080/00927872.2012.716120, URL https://doi.org/10.1080/00927872.2012.716120 Halverson and Ram [2005] Halverson T, Ram A (2005) Partition algebras. European J Combin 26(6):869–921. 10.1016/j.ejc.2004.06.005, URL http://dx.doi.org/10.1016/j.ejc.2004.06.005 Jacobson [1989] Jacobson N (1989) Basic algebra. II, 2nd edn. W. H. Freeman and Company, New York Jones [1994] Jones VFR (1994) The Potts model and the symmetric group. In: Subfactors (Kyuzeso, 1993). World Sci. Publ., River Edge, NJ, p 259–267 Kudryavtseva and Mazorchuk [2008] Kudryavtseva G, Mazorchuk V (2008) Schur-Weyl dualities for symmetric inverse semigroups. J Pure Appl Algebra 212(8):1987–1995. 10.1016/j.jpaa.2007.12.004, URL https://doi.org/10.1016/j.jpaa.2007.12.004 Lawson [1998] Lawson MV (1998) Inverse semigroups. World Scientific Publishing Co. Inc., River Edge, NJ, 10.1142/9789812816689, URL http://dx.doi.org/10.1142/9789812816689, the theory of partial symmetries Martin [1994] Martin P (1994) Temperley-Lieb algebras for nonplanar statistical mechanics—the partition algebra construction. J Knot Theory Ramifications 3(1):51–82. 10.1142/S0218216594000071, URL http://dx.doi.org/10.1142/S0218216594000071 Martin [1996] Martin P (1996) The structure of the partition algebras. J Algebra 183(2):319–358. 10.1006/jabr.1996.0223, URL http://dx.doi.org/10.1006/jabr.1996.0223 Martin and Mazorchuk [2014] Martin P, Mazorchuk V (2014) On the representation theory of partial Brauer algebras. Q J Math 65(1):225–247. 10.1093/qmath/has043, URL https://doi.org/10.1093/qmath/has043 Martin [1990] Martin PP (1990) Representations of graph Temperley-Lieb algebras. Publ Res Inst Math Sci 26(3):485–503. 10.2977/prims/1195170958, URL https://doi.org/10.2977/prims/1195170958 Martin [2008] Martin S (2008) Schur algebras and representation theory, Cambridge Tracts in Mathematics, vol 112. Cambridge University Press, Cambridge, reprint of the 1993 original Munn [1955] Munn WD (1955) On semigroup algebras. Proc Cambridge Philos Soc 51:1–15 Munn [1957a] Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Halverson T (2004) Representations of the q𝑞qitalic_q-rook monoid. J Algebra 273(1):227–251. 10.1016/j.jalgebra.2003.11.002, URL http://dx.doi.org/10.1016/j.jalgebra.2003.11.002 Halverson and delMas [2014] Halverson T, delMas E (2014) Representations of the Rook-Brauer algebra. Comm Algebra 42(1):423–443. 10.1080/00927872.2012.716120, URL https://doi.org/10.1080/00927872.2012.716120 Halverson and Ram [2005] Halverson T, Ram A (2005) Partition algebras. European J Combin 26(6):869–921. 10.1016/j.ejc.2004.06.005, URL http://dx.doi.org/10.1016/j.ejc.2004.06.005 Jacobson [1989] Jacobson N (1989) Basic algebra. II, 2nd edn. W. H. Freeman and Company, New York Jones [1994] Jones VFR (1994) The Potts model and the symmetric group. In: Subfactors (Kyuzeso, 1993). World Sci. Publ., River Edge, NJ, p 259–267 Kudryavtseva and Mazorchuk [2008] Kudryavtseva G, Mazorchuk V (2008) Schur-Weyl dualities for symmetric inverse semigroups. J Pure Appl Algebra 212(8):1987–1995. 10.1016/j.jpaa.2007.12.004, URL https://doi.org/10.1016/j.jpaa.2007.12.004 Lawson [1998] Lawson MV (1998) Inverse semigroups. World Scientific Publishing Co. Inc., River Edge, NJ, 10.1142/9789812816689, URL http://dx.doi.org/10.1142/9789812816689, the theory of partial symmetries Martin [1994] Martin P (1994) Temperley-Lieb algebras for nonplanar statistical mechanics—the partition algebra construction. J Knot Theory Ramifications 3(1):51–82. 10.1142/S0218216594000071, URL http://dx.doi.org/10.1142/S0218216594000071 Martin [1996] Martin P (1996) The structure of the partition algebras. J Algebra 183(2):319–358. 10.1006/jabr.1996.0223, URL http://dx.doi.org/10.1006/jabr.1996.0223 Martin and Mazorchuk [2014] Martin P, Mazorchuk V (2014) On the representation theory of partial Brauer algebras. Q J Math 65(1):225–247. 10.1093/qmath/has043, URL https://doi.org/10.1093/qmath/has043 Martin [1990] Martin PP (1990) Representations of graph Temperley-Lieb algebras. Publ Res Inst Math Sci 26(3):485–503. 10.2977/prims/1195170958, URL https://doi.org/10.2977/prims/1195170958 Martin [2008] Martin S (2008) Schur algebras and representation theory, Cambridge Tracts in Mathematics, vol 112. Cambridge University Press, Cambridge, reprint of the 1993 original Munn [1955] Munn WD (1955) On semigroup algebras. Proc Cambridge Philos Soc 51:1–15 Munn [1957a] Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Halverson T, delMas E (2014) Representations of the Rook-Brauer algebra. Comm Algebra 42(1):423–443. 10.1080/00927872.2012.716120, URL https://doi.org/10.1080/00927872.2012.716120 Halverson and Ram [2005] Halverson T, Ram A (2005) Partition algebras. European J Combin 26(6):869–921. 10.1016/j.ejc.2004.06.005, URL http://dx.doi.org/10.1016/j.ejc.2004.06.005 Jacobson [1989] Jacobson N (1989) Basic algebra. II, 2nd edn. W. H. Freeman and Company, New York Jones [1994] Jones VFR (1994) The Potts model and the symmetric group. In: Subfactors (Kyuzeso, 1993). World Sci. Publ., River Edge, NJ, p 259–267 Kudryavtseva and Mazorchuk [2008] Kudryavtseva G, Mazorchuk V (2008) Schur-Weyl dualities for symmetric inverse semigroups. J Pure Appl Algebra 212(8):1987–1995. 10.1016/j.jpaa.2007.12.004, URL https://doi.org/10.1016/j.jpaa.2007.12.004 Lawson [1998] Lawson MV (1998) Inverse semigroups. World Scientific Publishing Co. Inc., River Edge, NJ, 10.1142/9789812816689, URL http://dx.doi.org/10.1142/9789812816689, the theory of partial symmetries Martin [1994] Martin P (1994) Temperley-Lieb algebras for nonplanar statistical mechanics—the partition algebra construction. J Knot Theory Ramifications 3(1):51–82. 10.1142/S0218216594000071, URL http://dx.doi.org/10.1142/S0218216594000071 Martin [1996] Martin P (1996) The structure of the partition algebras. J Algebra 183(2):319–358. 10.1006/jabr.1996.0223, URL http://dx.doi.org/10.1006/jabr.1996.0223 Martin and Mazorchuk [2014] Martin P, Mazorchuk V (2014) On the representation theory of partial Brauer algebras. Q J Math 65(1):225–247. 10.1093/qmath/has043, URL https://doi.org/10.1093/qmath/has043 Martin [1990] Martin PP (1990) Representations of graph Temperley-Lieb algebras. Publ Res Inst Math Sci 26(3):485–503. 10.2977/prims/1195170958, URL https://doi.org/10.2977/prims/1195170958 Martin [2008] Martin S (2008) Schur algebras and representation theory, Cambridge Tracts in Mathematics, vol 112. Cambridge University Press, Cambridge, reprint of the 1993 original Munn [1955] Munn WD (1955) On semigroup algebras. Proc Cambridge Philos Soc 51:1–15 Munn [1957a] Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Halverson T, Ram A (2005) Partition algebras. European J Combin 26(6):869–921. 10.1016/j.ejc.2004.06.005, URL http://dx.doi.org/10.1016/j.ejc.2004.06.005 Jacobson [1989] Jacobson N (1989) Basic algebra. II, 2nd edn. W. H. Freeman and Company, New York Jones [1994] Jones VFR (1994) The Potts model and the symmetric group. In: Subfactors (Kyuzeso, 1993). World Sci. Publ., River Edge, NJ, p 259–267 Kudryavtseva and Mazorchuk [2008] Kudryavtseva G, Mazorchuk V (2008) Schur-Weyl dualities for symmetric inverse semigroups. J Pure Appl Algebra 212(8):1987–1995. 10.1016/j.jpaa.2007.12.004, URL https://doi.org/10.1016/j.jpaa.2007.12.004 Lawson [1998] Lawson MV (1998) Inverse semigroups. World Scientific Publishing Co. Inc., River Edge, NJ, 10.1142/9789812816689, URL http://dx.doi.org/10.1142/9789812816689, the theory of partial symmetries Martin [1994] Martin P (1994) Temperley-Lieb algebras for nonplanar statistical mechanics—the partition algebra construction. J Knot Theory Ramifications 3(1):51–82. 10.1142/S0218216594000071, URL http://dx.doi.org/10.1142/S0218216594000071 Martin [1996] Martin P (1996) The structure of the partition algebras. J Algebra 183(2):319–358. 10.1006/jabr.1996.0223, URL http://dx.doi.org/10.1006/jabr.1996.0223 Martin and Mazorchuk [2014] Martin P, Mazorchuk V (2014) On the representation theory of partial Brauer algebras. Q J Math 65(1):225–247. 10.1093/qmath/has043, URL https://doi.org/10.1093/qmath/has043 Martin [1990] Martin PP (1990) Representations of graph Temperley-Lieb algebras. Publ Res Inst Math Sci 26(3):485–503. 10.2977/prims/1195170958, URL https://doi.org/10.2977/prims/1195170958 Martin [2008] Martin S (2008) Schur algebras and representation theory, Cambridge Tracts in Mathematics, vol 112. Cambridge University Press, Cambridge, reprint of the 1993 original Munn [1955] Munn WD (1955) On semigroup algebras. Proc Cambridge Philos Soc 51:1–15 Munn [1957a] Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Jacobson N (1989) Basic algebra. II, 2nd edn. W. H. Freeman and Company, New York Jones [1994] Jones VFR (1994) The Potts model and the symmetric group. In: Subfactors (Kyuzeso, 1993). World Sci. Publ., River Edge, NJ, p 259–267 Kudryavtseva and Mazorchuk [2008] Kudryavtseva G, Mazorchuk V (2008) Schur-Weyl dualities for symmetric inverse semigroups. J Pure Appl Algebra 212(8):1987–1995. 10.1016/j.jpaa.2007.12.004, URL https://doi.org/10.1016/j.jpaa.2007.12.004 Lawson [1998] Lawson MV (1998) Inverse semigroups. World Scientific Publishing Co. Inc., River Edge, NJ, 10.1142/9789812816689, URL http://dx.doi.org/10.1142/9789812816689, the theory of partial symmetries Martin [1994] Martin P (1994) Temperley-Lieb algebras for nonplanar statistical mechanics—the partition algebra construction. J Knot Theory Ramifications 3(1):51–82. 10.1142/S0218216594000071, URL http://dx.doi.org/10.1142/S0218216594000071 Martin [1996] Martin P (1996) The structure of the partition algebras. J Algebra 183(2):319–358. 10.1006/jabr.1996.0223, URL http://dx.doi.org/10.1006/jabr.1996.0223 Martin and Mazorchuk [2014] Martin P, Mazorchuk V (2014) On the representation theory of partial Brauer algebras. Q J Math 65(1):225–247. 10.1093/qmath/has043, URL https://doi.org/10.1093/qmath/has043 Martin [1990] Martin PP (1990) Representations of graph Temperley-Lieb algebras. Publ Res Inst Math Sci 26(3):485–503. 10.2977/prims/1195170958, URL https://doi.org/10.2977/prims/1195170958 Martin [2008] Martin S (2008) Schur algebras and representation theory, Cambridge Tracts in Mathematics, vol 112. Cambridge University Press, Cambridge, reprint of the 1993 original Munn [1955] Munn WD (1955) On semigroup algebras. Proc Cambridge Philos Soc 51:1–15 Munn [1957a] Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Jones VFR (1994) The Potts model and the symmetric group. In: Subfactors (Kyuzeso, 1993). World Sci. Publ., River Edge, NJ, p 259–267 Kudryavtseva and Mazorchuk [2008] Kudryavtseva G, Mazorchuk V (2008) Schur-Weyl dualities for symmetric inverse semigroups. J Pure Appl Algebra 212(8):1987–1995. 10.1016/j.jpaa.2007.12.004, URL https://doi.org/10.1016/j.jpaa.2007.12.004 Lawson [1998] Lawson MV (1998) Inverse semigroups. World Scientific Publishing Co. Inc., River Edge, NJ, 10.1142/9789812816689, URL http://dx.doi.org/10.1142/9789812816689, the theory of partial symmetries Martin [1994] Martin P (1994) Temperley-Lieb algebras for nonplanar statistical mechanics—the partition algebra construction. J Knot Theory Ramifications 3(1):51–82. 10.1142/S0218216594000071, URL http://dx.doi.org/10.1142/S0218216594000071 Martin [1996] Martin P (1996) The structure of the partition algebras. J Algebra 183(2):319–358. 10.1006/jabr.1996.0223, URL http://dx.doi.org/10.1006/jabr.1996.0223 Martin and Mazorchuk [2014] Martin P, Mazorchuk V (2014) On the representation theory of partial Brauer algebras. Q J Math 65(1):225–247. 10.1093/qmath/has043, URL https://doi.org/10.1093/qmath/has043 Martin [1990] Martin PP (1990) Representations of graph Temperley-Lieb algebras. Publ Res Inst Math Sci 26(3):485–503. 10.2977/prims/1195170958, URL https://doi.org/10.2977/prims/1195170958 Martin [2008] Martin S (2008) Schur algebras and representation theory, Cambridge Tracts in Mathematics, vol 112. Cambridge University Press, Cambridge, reprint of the 1993 original Munn [1955] Munn WD (1955) On semigroup algebras. Proc Cambridge Philos Soc 51:1–15 Munn [1957a] Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Kudryavtseva G, Mazorchuk V (2008) Schur-Weyl dualities for symmetric inverse semigroups. J Pure Appl Algebra 212(8):1987–1995. 10.1016/j.jpaa.2007.12.004, URL https://doi.org/10.1016/j.jpaa.2007.12.004 Lawson [1998] Lawson MV (1998) Inverse semigroups. World Scientific Publishing Co. Inc., River Edge, NJ, 10.1142/9789812816689, URL http://dx.doi.org/10.1142/9789812816689, the theory of partial symmetries Martin [1994] Martin P (1994) Temperley-Lieb algebras for nonplanar statistical mechanics—the partition algebra construction. J Knot Theory Ramifications 3(1):51–82. 10.1142/S0218216594000071, URL http://dx.doi.org/10.1142/S0218216594000071 Martin [1996] Martin P (1996) The structure of the partition algebras. J Algebra 183(2):319–358. 10.1006/jabr.1996.0223, URL http://dx.doi.org/10.1006/jabr.1996.0223 Martin and Mazorchuk [2014] Martin P, Mazorchuk V (2014) On the representation theory of partial Brauer algebras. Q J Math 65(1):225–247. 10.1093/qmath/has043, URL https://doi.org/10.1093/qmath/has043 Martin [1990] Martin PP (1990) Representations of graph Temperley-Lieb algebras. Publ Res Inst Math Sci 26(3):485–503. 10.2977/prims/1195170958, URL https://doi.org/10.2977/prims/1195170958 Martin [2008] Martin S (2008) Schur algebras and representation theory, Cambridge Tracts in Mathematics, vol 112. Cambridge University Press, Cambridge, reprint of the 1993 original Munn [1955] Munn WD (1955) On semigroup algebras. Proc Cambridge Philos Soc 51:1–15 Munn [1957a] Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Lawson MV (1998) Inverse semigroups. World Scientific Publishing Co. Inc., River Edge, NJ, 10.1142/9789812816689, URL http://dx.doi.org/10.1142/9789812816689, the theory of partial symmetries Martin [1994] Martin P (1994) Temperley-Lieb algebras for nonplanar statistical mechanics—the partition algebra construction. J Knot Theory Ramifications 3(1):51–82. 10.1142/S0218216594000071, URL http://dx.doi.org/10.1142/S0218216594000071 Martin [1996] Martin P (1996) The structure of the partition algebras. J Algebra 183(2):319–358. 10.1006/jabr.1996.0223, URL http://dx.doi.org/10.1006/jabr.1996.0223 Martin and Mazorchuk [2014] Martin P, Mazorchuk V (2014) On the representation theory of partial Brauer algebras. Q J Math 65(1):225–247. 10.1093/qmath/has043, URL https://doi.org/10.1093/qmath/has043 Martin [1990] Martin PP (1990) Representations of graph Temperley-Lieb algebras. Publ Res Inst Math Sci 26(3):485–503. 10.2977/prims/1195170958, URL https://doi.org/10.2977/prims/1195170958 Martin [2008] Martin S (2008) Schur algebras and representation theory, Cambridge Tracts in Mathematics, vol 112. Cambridge University Press, Cambridge, reprint of the 1993 original Munn [1955] Munn WD (1955) On semigroup algebras. Proc Cambridge Philos Soc 51:1–15 Munn [1957a] Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Martin P (1994) Temperley-Lieb algebras for nonplanar statistical mechanics—the partition algebra construction. J Knot Theory Ramifications 3(1):51–82. 10.1142/S0218216594000071, URL http://dx.doi.org/10.1142/S0218216594000071 Martin [1996] Martin P (1996) The structure of the partition algebras. J Algebra 183(2):319–358. 10.1006/jabr.1996.0223, URL http://dx.doi.org/10.1006/jabr.1996.0223 Martin and Mazorchuk [2014] Martin P, Mazorchuk V (2014) On the representation theory of partial Brauer algebras. Q J Math 65(1):225–247. 10.1093/qmath/has043, URL https://doi.org/10.1093/qmath/has043 Martin [1990] Martin PP (1990) Representations of graph Temperley-Lieb algebras. Publ Res Inst Math Sci 26(3):485–503. 10.2977/prims/1195170958, URL https://doi.org/10.2977/prims/1195170958 Martin [2008] Martin S (2008) Schur algebras and representation theory, Cambridge Tracts in Mathematics, vol 112. Cambridge University Press, Cambridge, reprint of the 1993 original Munn [1955] Munn WD (1955) On semigroup algebras. Proc Cambridge Philos Soc 51:1–15 Munn [1957a] Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Martin P (1996) The structure of the partition algebras. J Algebra 183(2):319–358. 10.1006/jabr.1996.0223, URL http://dx.doi.org/10.1006/jabr.1996.0223 Martin and Mazorchuk [2014] Martin P, Mazorchuk V (2014) On the representation theory of partial Brauer algebras. Q J Math 65(1):225–247. 10.1093/qmath/has043, URL https://doi.org/10.1093/qmath/has043 Martin [1990] Martin PP (1990) Representations of graph Temperley-Lieb algebras. Publ Res Inst Math Sci 26(3):485–503. 10.2977/prims/1195170958, URL https://doi.org/10.2977/prims/1195170958 Martin [2008] Martin S (2008) Schur algebras and representation theory, Cambridge Tracts in Mathematics, vol 112. Cambridge University Press, Cambridge, reprint of the 1993 original Munn [1955] Munn WD (1955) On semigroup algebras. Proc Cambridge Philos Soc 51:1–15 Munn [1957a] Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Martin P, Mazorchuk V (2014) On the representation theory of partial Brauer algebras. Q J Math 65(1):225–247. 10.1093/qmath/has043, URL https://doi.org/10.1093/qmath/has043 Martin [1990] Martin PP (1990) Representations of graph Temperley-Lieb algebras. Publ Res Inst Math Sci 26(3):485–503. 10.2977/prims/1195170958, URL https://doi.org/10.2977/prims/1195170958 Martin [2008] Martin S (2008) Schur algebras and representation theory, Cambridge Tracts in Mathematics, vol 112. Cambridge University Press, Cambridge, reprint of the 1993 original Munn [1955] Munn WD (1955) On semigroup algebras. Proc Cambridge Philos Soc 51:1–15 Munn [1957a] Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Martin PP (1990) Representations of graph Temperley-Lieb algebras. Publ Res Inst Math Sci 26(3):485–503. 10.2977/prims/1195170958, URL https://doi.org/10.2977/prims/1195170958 Martin [2008] Martin S (2008) Schur algebras and representation theory, Cambridge Tracts in Mathematics, vol 112. Cambridge University Press, Cambridge, reprint of the 1993 original Munn [1955] Munn WD (1955) On semigroup algebras. Proc Cambridge Philos Soc 51:1–15 Munn [1957a] Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Martin S (2008) Schur algebras and representation theory, Cambridge Tracts in Mathematics, vol 112. Cambridge University Press, Cambridge, reprint of the 1993 original Munn [1955] Munn WD (1955) On semigroup algebras. Proc Cambridge Philos Soc 51:1–15 Munn [1957a] Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Munn WD (1955) On semigroup algebras. Proc Cambridge Philos Soc 51:1–15 Munn [1957a] Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. 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Springer, Dordrecht, 10.1007/978-0-387-79852-3, URL https://doi.org/10.1007/978-0-387-79852-3 Graham and Lehrer [1996] Graham JJ, Lehrer GI (1996) Cellular algebras. Invent Math 123(1):1–34. 10.1007/BF01232365, URL http://dx.doi.org/10.1007/BF01232365 Green [1980] Green JA (1980) Polynomial representations of GLnsubscriptGL𝑛{\rm GL}_{n}roman_GL start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT, Lecture Notes in Mathematics, vol 830. Springer-Verlag, Berlin Grood [2006] Grood C (2006) The rook partition algebra. J Combin Theory Ser A 113(2):325–351. 10.1016/j.jcta.2005.03.006, URL https://doi.org/10.1016/j.jcta.2005.03.006 Halverson [1996] Halverson T (1996) Characters of the centralizer algebras of mixed tensor representations of Gl(r,ℂ)Gl𝑟ℂ{\rm Gl}(r,\mathbb{C})roman_Gl ( italic_r , blackboard_C ) and the quantum group 𝔘q(ℊl(r,ℂ))subscript𝔘𝑞ℊ𝑙𝑟ℂ\mathfrak{U}_{q}({\mathcal{g}l}(r,\mathbb{C}))fraktur_U start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ( caligraphic_g italic_l ( italic_r , blackboard_C ) ). Pacific J Math 174(2):359–410. URL http://projecteuclid.org/euclid.pjm/1102365176 Halverson [2004] Halverson T (2004) Representations of the q𝑞qitalic_q-rook monoid. J Algebra 273(1):227–251. 10.1016/j.jalgebra.2003.11.002, URL http://dx.doi.org/10.1016/j.jalgebra.2003.11.002 Halverson and delMas [2014] Halverson T, delMas E (2014) Representations of the Rook-Brauer algebra. Comm Algebra 42(1):423–443. 10.1080/00927872.2012.716120, URL https://doi.org/10.1080/00927872.2012.716120 Halverson and Ram [2005] Halverson T, Ram A (2005) Partition algebras. European J Combin 26(6):869–921. 10.1016/j.ejc.2004.06.005, URL http://dx.doi.org/10.1016/j.ejc.2004.06.005 Jacobson [1989] Jacobson N (1989) Basic algebra. II, 2nd edn. W. H. Freeman and Company, New York Jones [1994] Jones VFR (1994) The Potts model and the symmetric group. In: Subfactors (Kyuzeso, 1993). World Sci. Publ., River Edge, NJ, p 259–267 Kudryavtseva and Mazorchuk [2008] Kudryavtseva G, Mazorchuk V (2008) Schur-Weyl dualities for symmetric inverse semigroups. J Pure Appl Algebra 212(8):1987–1995. 10.1016/j.jpaa.2007.12.004, URL https://doi.org/10.1016/j.jpaa.2007.12.004 Lawson [1998] Lawson MV (1998) Inverse semigroups. World Scientific Publishing Co. Inc., River Edge, NJ, 10.1142/9789812816689, URL http://dx.doi.org/10.1142/9789812816689, the theory of partial symmetries Martin [1994] Martin P (1994) Temperley-Lieb algebras for nonplanar statistical mechanics—the partition algebra construction. J Knot Theory Ramifications 3(1):51–82. 10.1142/S0218216594000071, URL http://dx.doi.org/10.1142/S0218216594000071 Martin [1996] Martin P (1996) The structure of the partition algebras. J Algebra 183(2):319–358. 10.1006/jabr.1996.0223, URL http://dx.doi.org/10.1006/jabr.1996.0223 Martin and Mazorchuk [2014] Martin P, Mazorchuk V (2014) On the representation theory of partial Brauer algebras. Q J Math 65(1):225–247. 10.1093/qmath/has043, URL https://doi.org/10.1093/qmath/has043 Martin [1990] Martin PP (1990) Representations of graph Temperley-Lieb algebras. Publ Res Inst Math Sci 26(3):485–503. 10.2977/prims/1195170958, URL https://doi.org/10.2977/prims/1195170958 Martin [2008] Martin S (2008) Schur algebras and representation theory, Cambridge Tracts in Mathematics, vol 112. Cambridge University Press, Cambridge, reprint of the 1993 original Munn [1955] Munn WD (1955) On semigroup algebras. Proc Cambridge Philos Soc 51:1–15 Munn [1957a] Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Dipper R, Doty S, Hu J (2008) Brauer algebras, symplectic Schur algebras and Schur-Weyl duality. Trans Amer Math Soc 360(1):189–213. 10.1090/S0002-9947-07-04179-7, URL https://doi.org/10.1090/S0002-9947-07-04179-7 Doty and Hu [2009] Doty S, Hu J (2009) Schur-Weyl duality for orthogonal groups. Proc Lond Math Soc (3) 98(3):679–713. 10.1112/plms/pdn044, URL https://doi.org/10.1112/plms/pdn044 Goodman and Wallach [2009] Goodman R, Wallach NR (2009) Symmetry, representations, and invariants, Graduate Texts in Mathematics, vol 255. Springer, Dordrecht, 10.1007/978-0-387-79852-3, URL https://doi.org/10.1007/978-0-387-79852-3 Graham and Lehrer [1996] Graham JJ, Lehrer GI (1996) Cellular algebras. Invent Math 123(1):1–34. 10.1007/BF01232365, URL http://dx.doi.org/10.1007/BF01232365 Green [1980] Green JA (1980) Polynomial representations of GLnsubscriptGL𝑛{\rm GL}_{n}roman_GL start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT, Lecture Notes in Mathematics, vol 830. Springer-Verlag, Berlin Grood [2006] Grood C (2006) The rook partition algebra. J Combin Theory Ser A 113(2):325–351. 10.1016/j.jcta.2005.03.006, URL https://doi.org/10.1016/j.jcta.2005.03.006 Halverson [1996] Halverson T (1996) Characters of the centralizer algebras of mixed tensor representations of Gl(r,ℂ)Gl𝑟ℂ{\rm Gl}(r,\mathbb{C})roman_Gl ( italic_r , blackboard_C ) and the quantum group 𝔘q(ℊl(r,ℂ))subscript𝔘𝑞ℊ𝑙𝑟ℂ\mathfrak{U}_{q}({\mathcal{g}l}(r,\mathbb{C}))fraktur_U start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ( caligraphic_g italic_l ( italic_r , blackboard_C ) ). Pacific J Math 174(2):359–410. URL http://projecteuclid.org/euclid.pjm/1102365176 Halverson [2004] Halverson T (2004) Representations of the q𝑞qitalic_q-rook monoid. J Algebra 273(1):227–251. 10.1016/j.jalgebra.2003.11.002, URL http://dx.doi.org/10.1016/j.jalgebra.2003.11.002 Halverson and delMas [2014] Halverson T, delMas E (2014) Representations of the Rook-Brauer algebra. Comm Algebra 42(1):423–443. 10.1080/00927872.2012.716120, URL https://doi.org/10.1080/00927872.2012.716120 Halverson and Ram [2005] Halverson T, Ram A (2005) Partition algebras. European J Combin 26(6):869–921. 10.1016/j.ejc.2004.06.005, URL http://dx.doi.org/10.1016/j.ejc.2004.06.005 Jacobson [1989] Jacobson N (1989) Basic algebra. II, 2nd edn. W. H. Freeman and Company, New York Jones [1994] Jones VFR (1994) The Potts model and the symmetric group. In: Subfactors (Kyuzeso, 1993). World Sci. Publ., River Edge, NJ, p 259–267 Kudryavtseva and Mazorchuk [2008] Kudryavtseva G, Mazorchuk V (2008) Schur-Weyl dualities for symmetric inverse semigroups. J Pure Appl Algebra 212(8):1987–1995. 10.1016/j.jpaa.2007.12.004, URL https://doi.org/10.1016/j.jpaa.2007.12.004 Lawson [1998] Lawson MV (1998) Inverse semigroups. World Scientific Publishing Co. Inc., River Edge, NJ, 10.1142/9789812816689, URL http://dx.doi.org/10.1142/9789812816689, the theory of partial symmetries Martin [1994] Martin P (1994) Temperley-Lieb algebras for nonplanar statistical mechanics—the partition algebra construction. J Knot Theory Ramifications 3(1):51–82. 10.1142/S0218216594000071, URL http://dx.doi.org/10.1142/S0218216594000071 Martin [1996] Martin P (1996) The structure of the partition algebras. J Algebra 183(2):319–358. 10.1006/jabr.1996.0223, URL http://dx.doi.org/10.1006/jabr.1996.0223 Martin and Mazorchuk [2014] Martin P, Mazorchuk V (2014) On the representation theory of partial Brauer algebras. Q J Math 65(1):225–247. 10.1093/qmath/has043, URL https://doi.org/10.1093/qmath/has043 Martin [1990] Martin PP (1990) Representations of graph Temperley-Lieb algebras. Publ Res Inst Math Sci 26(3):485–503. 10.2977/prims/1195170958, URL https://doi.org/10.2977/prims/1195170958 Martin [2008] Martin S (2008) Schur algebras and representation theory, Cambridge Tracts in Mathematics, vol 112. Cambridge University Press, Cambridge, reprint of the 1993 original Munn [1955] Munn WD (1955) On semigroup algebras. Proc Cambridge Philos Soc 51:1–15 Munn [1957a] Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Doty S, Hu J (2009) Schur-Weyl duality for orthogonal groups. Proc Lond Math Soc (3) 98(3):679–713. 10.1112/plms/pdn044, URL https://doi.org/10.1112/plms/pdn044 Goodman and Wallach [2009] Goodman R, Wallach NR (2009) Symmetry, representations, and invariants, Graduate Texts in Mathematics, vol 255. Springer, Dordrecht, 10.1007/978-0-387-79852-3, URL https://doi.org/10.1007/978-0-387-79852-3 Graham and Lehrer [1996] Graham JJ, Lehrer GI (1996) Cellular algebras. Invent Math 123(1):1–34. 10.1007/BF01232365, URL http://dx.doi.org/10.1007/BF01232365 Green [1980] Green JA (1980) Polynomial representations of GLnsubscriptGL𝑛{\rm GL}_{n}roman_GL start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT, Lecture Notes in Mathematics, vol 830. Springer-Verlag, Berlin Grood [2006] Grood C (2006) The rook partition algebra. J Combin Theory Ser A 113(2):325–351. 10.1016/j.jcta.2005.03.006, URL https://doi.org/10.1016/j.jcta.2005.03.006 Halverson [1996] Halverson T (1996) Characters of the centralizer algebras of mixed tensor representations of Gl(r,ℂ)Gl𝑟ℂ{\rm Gl}(r,\mathbb{C})roman_Gl ( italic_r , blackboard_C ) and the quantum group 𝔘q(ℊl(r,ℂ))subscript𝔘𝑞ℊ𝑙𝑟ℂ\mathfrak{U}_{q}({\mathcal{g}l}(r,\mathbb{C}))fraktur_U start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ( caligraphic_g italic_l ( italic_r , blackboard_C ) ). Pacific J Math 174(2):359–410. URL http://projecteuclid.org/euclid.pjm/1102365176 Halverson [2004] Halverson T (2004) Representations of the q𝑞qitalic_q-rook monoid. J Algebra 273(1):227–251. 10.1016/j.jalgebra.2003.11.002, URL http://dx.doi.org/10.1016/j.jalgebra.2003.11.002 Halverson and delMas [2014] Halverson T, delMas E (2014) Representations of the Rook-Brauer algebra. Comm Algebra 42(1):423–443. 10.1080/00927872.2012.716120, URL https://doi.org/10.1080/00927872.2012.716120 Halverson and Ram [2005] Halverson T, Ram A (2005) Partition algebras. European J Combin 26(6):869–921. 10.1016/j.ejc.2004.06.005, URL http://dx.doi.org/10.1016/j.ejc.2004.06.005 Jacobson [1989] Jacobson N (1989) Basic algebra. II, 2nd edn. W. H. Freeman and Company, New York Jones [1994] Jones VFR (1994) The Potts model and the symmetric group. In: Subfactors (Kyuzeso, 1993). World Sci. Publ., River Edge, NJ, p 259–267 Kudryavtseva and Mazorchuk [2008] Kudryavtseva G, Mazorchuk V (2008) Schur-Weyl dualities for symmetric inverse semigroups. J Pure Appl Algebra 212(8):1987–1995. 10.1016/j.jpaa.2007.12.004, URL https://doi.org/10.1016/j.jpaa.2007.12.004 Lawson [1998] Lawson MV (1998) Inverse semigroups. World Scientific Publishing Co. Inc., River Edge, NJ, 10.1142/9789812816689, URL http://dx.doi.org/10.1142/9789812816689, the theory of partial symmetries Martin [1994] Martin P (1994) Temperley-Lieb algebras for nonplanar statistical mechanics—the partition algebra construction. J Knot Theory Ramifications 3(1):51–82. 10.1142/S0218216594000071, URL http://dx.doi.org/10.1142/S0218216594000071 Martin [1996] Martin P (1996) The structure of the partition algebras. J Algebra 183(2):319–358. 10.1006/jabr.1996.0223, URL http://dx.doi.org/10.1006/jabr.1996.0223 Martin and Mazorchuk [2014] Martin P, Mazorchuk V (2014) On the representation theory of partial Brauer algebras. Q J Math 65(1):225–247. 10.1093/qmath/has043, URL https://doi.org/10.1093/qmath/has043 Martin [1990] Martin PP (1990) Representations of graph Temperley-Lieb algebras. Publ Res Inst Math Sci 26(3):485–503. 10.2977/prims/1195170958, URL https://doi.org/10.2977/prims/1195170958 Martin [2008] Martin S (2008) Schur algebras and representation theory, Cambridge Tracts in Mathematics, vol 112. Cambridge University Press, Cambridge, reprint of the 1993 original Munn [1955] Munn WD (1955) On semigroup algebras. Proc Cambridge Philos Soc 51:1–15 Munn [1957a] Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Goodman R, Wallach NR (2009) Symmetry, representations, and invariants, Graduate Texts in Mathematics, vol 255. Springer, Dordrecht, 10.1007/978-0-387-79852-3, URL https://doi.org/10.1007/978-0-387-79852-3 Graham and Lehrer [1996] Graham JJ, Lehrer GI (1996) Cellular algebras. Invent Math 123(1):1–34. 10.1007/BF01232365, URL http://dx.doi.org/10.1007/BF01232365 Green [1980] Green JA (1980) Polynomial representations of GLnsubscriptGL𝑛{\rm GL}_{n}roman_GL start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT, Lecture Notes in Mathematics, vol 830. Springer-Verlag, Berlin Grood [2006] Grood C (2006) The rook partition algebra. J Combin Theory Ser A 113(2):325–351. 10.1016/j.jcta.2005.03.006, URL https://doi.org/10.1016/j.jcta.2005.03.006 Halverson [1996] Halverson T (1996) Characters of the centralizer algebras of mixed tensor representations of Gl(r,ℂ)Gl𝑟ℂ{\rm Gl}(r,\mathbb{C})roman_Gl ( italic_r , blackboard_C ) and the quantum group 𝔘q(ℊl(r,ℂ))subscript𝔘𝑞ℊ𝑙𝑟ℂ\mathfrak{U}_{q}({\mathcal{g}l}(r,\mathbb{C}))fraktur_U start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ( caligraphic_g italic_l ( italic_r , blackboard_C ) ). Pacific J Math 174(2):359–410. URL http://projecteuclid.org/euclid.pjm/1102365176 Halverson [2004] Halverson T (2004) Representations of the q𝑞qitalic_q-rook monoid. J Algebra 273(1):227–251. 10.1016/j.jalgebra.2003.11.002, URL http://dx.doi.org/10.1016/j.jalgebra.2003.11.002 Halverson and delMas [2014] Halverson T, delMas E (2014) Representations of the Rook-Brauer algebra. Comm Algebra 42(1):423–443. 10.1080/00927872.2012.716120, URL https://doi.org/10.1080/00927872.2012.716120 Halverson and Ram [2005] Halverson T, Ram A (2005) Partition algebras. European J Combin 26(6):869–921. 10.1016/j.ejc.2004.06.005, URL http://dx.doi.org/10.1016/j.ejc.2004.06.005 Jacobson [1989] Jacobson N (1989) Basic algebra. II, 2nd edn. W. H. Freeman and Company, New York Jones [1994] Jones VFR (1994) The Potts model and the symmetric group. In: Subfactors (Kyuzeso, 1993). World Sci. Publ., River Edge, NJ, p 259–267 Kudryavtseva and Mazorchuk [2008] Kudryavtseva G, Mazorchuk V (2008) Schur-Weyl dualities for symmetric inverse semigroups. J Pure Appl Algebra 212(8):1987–1995. 10.1016/j.jpaa.2007.12.004, URL https://doi.org/10.1016/j.jpaa.2007.12.004 Lawson [1998] Lawson MV (1998) Inverse semigroups. World Scientific Publishing Co. Inc., River Edge, NJ, 10.1142/9789812816689, URL http://dx.doi.org/10.1142/9789812816689, the theory of partial symmetries Martin [1994] Martin P (1994) Temperley-Lieb algebras for nonplanar statistical mechanics—the partition algebra construction. J Knot Theory Ramifications 3(1):51–82. 10.1142/S0218216594000071, URL http://dx.doi.org/10.1142/S0218216594000071 Martin [1996] Martin P (1996) The structure of the partition algebras. J Algebra 183(2):319–358. 10.1006/jabr.1996.0223, URL http://dx.doi.org/10.1006/jabr.1996.0223 Martin and Mazorchuk [2014] Martin P, Mazorchuk V (2014) On the representation theory of partial Brauer algebras. Q J Math 65(1):225–247. 10.1093/qmath/has043, URL https://doi.org/10.1093/qmath/has043 Martin [1990] Martin PP (1990) Representations of graph Temperley-Lieb algebras. Publ Res Inst Math Sci 26(3):485–503. 10.2977/prims/1195170958, URL https://doi.org/10.2977/prims/1195170958 Martin [2008] Martin S (2008) Schur algebras and representation theory, Cambridge Tracts in Mathematics, vol 112. Cambridge University Press, Cambridge, reprint of the 1993 original Munn [1955] Munn WD (1955) On semigroup algebras. Proc Cambridge Philos Soc 51:1–15 Munn [1957a] Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Graham JJ, Lehrer GI (1996) Cellular algebras. Invent Math 123(1):1–34. 10.1007/BF01232365, URL http://dx.doi.org/10.1007/BF01232365 Green [1980] Green JA (1980) Polynomial representations of GLnsubscriptGL𝑛{\rm GL}_{n}roman_GL start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT, Lecture Notes in Mathematics, vol 830. Springer-Verlag, Berlin Grood [2006] Grood C (2006) The rook partition algebra. J Combin Theory Ser A 113(2):325–351. 10.1016/j.jcta.2005.03.006, URL https://doi.org/10.1016/j.jcta.2005.03.006 Halverson [1996] Halverson T (1996) Characters of the centralizer algebras of mixed tensor representations of Gl(r,ℂ)Gl𝑟ℂ{\rm Gl}(r,\mathbb{C})roman_Gl ( italic_r , blackboard_C ) and the quantum group 𝔘q(ℊl(r,ℂ))subscript𝔘𝑞ℊ𝑙𝑟ℂ\mathfrak{U}_{q}({\mathcal{g}l}(r,\mathbb{C}))fraktur_U start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ( caligraphic_g italic_l ( italic_r , blackboard_C ) ). Pacific J Math 174(2):359–410. URL http://projecteuclid.org/euclid.pjm/1102365176 Halverson [2004] Halverson T (2004) Representations of the q𝑞qitalic_q-rook monoid. J Algebra 273(1):227–251. 10.1016/j.jalgebra.2003.11.002, URL http://dx.doi.org/10.1016/j.jalgebra.2003.11.002 Halverson and delMas [2014] Halverson T, delMas E (2014) Representations of the Rook-Brauer algebra. Comm Algebra 42(1):423–443. 10.1080/00927872.2012.716120, URL https://doi.org/10.1080/00927872.2012.716120 Halverson and Ram [2005] Halverson T, Ram A (2005) Partition algebras. European J Combin 26(6):869–921. 10.1016/j.ejc.2004.06.005, URL http://dx.doi.org/10.1016/j.ejc.2004.06.005 Jacobson [1989] Jacobson N (1989) Basic algebra. II, 2nd edn. W. H. Freeman and Company, New York Jones [1994] Jones VFR (1994) The Potts model and the symmetric group. In: Subfactors (Kyuzeso, 1993). World Sci. Publ., River Edge, NJ, p 259–267 Kudryavtseva and Mazorchuk [2008] Kudryavtseva G, Mazorchuk V (2008) Schur-Weyl dualities for symmetric inverse semigroups. J Pure Appl Algebra 212(8):1987–1995. 10.1016/j.jpaa.2007.12.004, URL https://doi.org/10.1016/j.jpaa.2007.12.004 Lawson [1998] Lawson MV (1998) Inverse semigroups. World Scientific Publishing Co. Inc., River Edge, NJ, 10.1142/9789812816689, URL http://dx.doi.org/10.1142/9789812816689, the theory of partial symmetries Martin [1994] Martin P (1994) Temperley-Lieb algebras for nonplanar statistical mechanics—the partition algebra construction. J Knot Theory Ramifications 3(1):51–82. 10.1142/S0218216594000071, URL http://dx.doi.org/10.1142/S0218216594000071 Martin [1996] Martin P (1996) The structure of the partition algebras. J Algebra 183(2):319–358. 10.1006/jabr.1996.0223, URL http://dx.doi.org/10.1006/jabr.1996.0223 Martin and Mazorchuk [2014] Martin P, Mazorchuk V (2014) On the representation theory of partial Brauer algebras. Q J Math 65(1):225–247. 10.1093/qmath/has043, URL https://doi.org/10.1093/qmath/has043 Martin [1990] Martin PP (1990) Representations of graph Temperley-Lieb algebras. Publ Res Inst Math Sci 26(3):485–503. 10.2977/prims/1195170958, URL https://doi.org/10.2977/prims/1195170958 Martin [2008] Martin S (2008) Schur algebras and representation theory, Cambridge Tracts in Mathematics, vol 112. Cambridge University Press, Cambridge, reprint of the 1993 original Munn [1955] Munn WD (1955) On semigroup algebras. Proc Cambridge Philos Soc 51:1–15 Munn [1957a] Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Green JA (1980) Polynomial representations of GLnsubscriptGL𝑛{\rm GL}_{n}roman_GL start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT, Lecture Notes in Mathematics, vol 830. Springer-Verlag, Berlin Grood [2006] Grood C (2006) The rook partition algebra. J Combin Theory Ser A 113(2):325–351. 10.1016/j.jcta.2005.03.006, URL https://doi.org/10.1016/j.jcta.2005.03.006 Halverson [1996] Halverson T (1996) Characters of the centralizer algebras of mixed tensor representations of Gl(r,ℂ)Gl𝑟ℂ{\rm Gl}(r,\mathbb{C})roman_Gl ( italic_r , blackboard_C ) and the quantum group 𝔘q(ℊl(r,ℂ))subscript𝔘𝑞ℊ𝑙𝑟ℂ\mathfrak{U}_{q}({\mathcal{g}l}(r,\mathbb{C}))fraktur_U start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ( caligraphic_g italic_l ( italic_r , blackboard_C ) ). Pacific J Math 174(2):359–410. URL http://projecteuclid.org/euclid.pjm/1102365176 Halverson [2004] Halverson T (2004) Representations of the q𝑞qitalic_q-rook monoid. J Algebra 273(1):227–251. 10.1016/j.jalgebra.2003.11.002, URL http://dx.doi.org/10.1016/j.jalgebra.2003.11.002 Halverson and delMas [2014] Halverson T, delMas E (2014) Representations of the Rook-Brauer algebra. Comm Algebra 42(1):423–443. 10.1080/00927872.2012.716120, URL https://doi.org/10.1080/00927872.2012.716120 Halverson and Ram [2005] Halverson T, Ram A (2005) Partition algebras. European J Combin 26(6):869–921. 10.1016/j.ejc.2004.06.005, URL http://dx.doi.org/10.1016/j.ejc.2004.06.005 Jacobson [1989] Jacobson N (1989) Basic algebra. II, 2nd edn. W. H. Freeman and Company, New York Jones [1994] Jones VFR (1994) The Potts model and the symmetric group. In: Subfactors (Kyuzeso, 1993). World Sci. Publ., River Edge, NJ, p 259–267 Kudryavtseva and Mazorchuk [2008] Kudryavtseva G, Mazorchuk V (2008) Schur-Weyl dualities for symmetric inverse semigroups. J Pure Appl Algebra 212(8):1987–1995. 10.1016/j.jpaa.2007.12.004, URL https://doi.org/10.1016/j.jpaa.2007.12.004 Lawson [1998] Lawson MV (1998) Inverse semigroups. World Scientific Publishing Co. Inc., River Edge, NJ, 10.1142/9789812816689, URL http://dx.doi.org/10.1142/9789812816689, the theory of partial symmetries Martin [1994] Martin P (1994) Temperley-Lieb algebras for nonplanar statistical mechanics—the partition algebra construction. J Knot Theory Ramifications 3(1):51–82. 10.1142/S0218216594000071, URL http://dx.doi.org/10.1142/S0218216594000071 Martin [1996] Martin P (1996) The structure of the partition algebras. J Algebra 183(2):319–358. 10.1006/jabr.1996.0223, URL http://dx.doi.org/10.1006/jabr.1996.0223 Martin and Mazorchuk [2014] Martin P, Mazorchuk V (2014) On the representation theory of partial Brauer algebras. Q J Math 65(1):225–247. 10.1093/qmath/has043, URL https://doi.org/10.1093/qmath/has043 Martin [1990] Martin PP (1990) Representations of graph Temperley-Lieb algebras. Publ Res Inst Math Sci 26(3):485–503. 10.2977/prims/1195170958, URL https://doi.org/10.2977/prims/1195170958 Martin [2008] Martin S (2008) Schur algebras and representation theory, Cambridge Tracts in Mathematics, vol 112. Cambridge University Press, Cambridge, reprint of the 1993 original Munn [1955] Munn WD (1955) On semigroup algebras. Proc Cambridge Philos Soc 51:1–15 Munn [1957a] Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Grood C (2006) The rook partition algebra. J Combin Theory Ser A 113(2):325–351. 10.1016/j.jcta.2005.03.006, URL https://doi.org/10.1016/j.jcta.2005.03.006 Halverson [1996] Halverson T (1996) Characters of the centralizer algebras of mixed tensor representations of Gl(r,ℂ)Gl𝑟ℂ{\rm Gl}(r,\mathbb{C})roman_Gl ( italic_r , blackboard_C ) and the quantum group 𝔘q(ℊl(r,ℂ))subscript𝔘𝑞ℊ𝑙𝑟ℂ\mathfrak{U}_{q}({\mathcal{g}l}(r,\mathbb{C}))fraktur_U start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ( caligraphic_g italic_l ( italic_r , blackboard_C ) ). Pacific J Math 174(2):359–410. URL http://projecteuclid.org/euclid.pjm/1102365176 Halverson [2004] Halverson T (2004) Representations of the q𝑞qitalic_q-rook monoid. J Algebra 273(1):227–251. 10.1016/j.jalgebra.2003.11.002, URL http://dx.doi.org/10.1016/j.jalgebra.2003.11.002 Halverson and delMas [2014] Halverson T, delMas E (2014) Representations of the Rook-Brauer algebra. Comm Algebra 42(1):423–443. 10.1080/00927872.2012.716120, URL https://doi.org/10.1080/00927872.2012.716120 Halverson and Ram [2005] Halverson T, Ram A (2005) Partition algebras. European J Combin 26(6):869–921. 10.1016/j.ejc.2004.06.005, URL http://dx.doi.org/10.1016/j.ejc.2004.06.005 Jacobson [1989] Jacobson N (1989) Basic algebra. II, 2nd edn. W. H. Freeman and Company, New York Jones [1994] Jones VFR (1994) The Potts model and the symmetric group. In: Subfactors (Kyuzeso, 1993). World Sci. Publ., River Edge, NJ, p 259–267 Kudryavtseva and Mazorchuk [2008] Kudryavtseva G, Mazorchuk V (2008) Schur-Weyl dualities for symmetric inverse semigroups. J Pure Appl Algebra 212(8):1987–1995. 10.1016/j.jpaa.2007.12.004, URL https://doi.org/10.1016/j.jpaa.2007.12.004 Lawson [1998] Lawson MV (1998) Inverse semigroups. World Scientific Publishing Co. Inc., River Edge, NJ, 10.1142/9789812816689, URL http://dx.doi.org/10.1142/9789812816689, the theory of partial symmetries Martin [1994] Martin P (1994) Temperley-Lieb algebras for nonplanar statistical mechanics—the partition algebra construction. J Knot Theory Ramifications 3(1):51–82. 10.1142/S0218216594000071, URL http://dx.doi.org/10.1142/S0218216594000071 Martin [1996] Martin P (1996) The structure of the partition algebras. J Algebra 183(2):319–358. 10.1006/jabr.1996.0223, URL http://dx.doi.org/10.1006/jabr.1996.0223 Martin and Mazorchuk [2014] Martin P, Mazorchuk V (2014) On the representation theory of partial Brauer algebras. Q J Math 65(1):225–247. 10.1093/qmath/has043, URL https://doi.org/10.1093/qmath/has043 Martin [1990] Martin PP (1990) Representations of graph Temperley-Lieb algebras. Publ Res Inst Math Sci 26(3):485–503. 10.2977/prims/1195170958, URL https://doi.org/10.2977/prims/1195170958 Martin [2008] Martin S (2008) Schur algebras and representation theory, Cambridge Tracts in Mathematics, vol 112. Cambridge University Press, Cambridge, reprint of the 1993 original Munn [1955] Munn WD (1955) On semigroup algebras. Proc Cambridge Philos Soc 51:1–15 Munn [1957a] Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Halverson T (1996) Characters of the centralizer algebras of mixed tensor representations of Gl(r,ℂ)Gl𝑟ℂ{\rm Gl}(r,\mathbb{C})roman_Gl ( italic_r , blackboard_C ) and the quantum group 𝔘q(ℊl(r,ℂ))subscript𝔘𝑞ℊ𝑙𝑟ℂ\mathfrak{U}_{q}({\mathcal{g}l}(r,\mathbb{C}))fraktur_U start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ( caligraphic_g italic_l ( italic_r , blackboard_C ) ). Pacific J Math 174(2):359–410. URL http://projecteuclid.org/euclid.pjm/1102365176 Halverson [2004] Halverson T (2004) Representations of the q𝑞qitalic_q-rook monoid. J Algebra 273(1):227–251. 10.1016/j.jalgebra.2003.11.002, URL http://dx.doi.org/10.1016/j.jalgebra.2003.11.002 Halverson and delMas [2014] Halverson T, delMas E (2014) Representations of the Rook-Brauer algebra. Comm Algebra 42(1):423–443. 10.1080/00927872.2012.716120, URL https://doi.org/10.1080/00927872.2012.716120 Halverson and Ram [2005] Halverson T, Ram A (2005) Partition algebras. European J Combin 26(6):869–921. 10.1016/j.ejc.2004.06.005, URL http://dx.doi.org/10.1016/j.ejc.2004.06.005 Jacobson [1989] Jacobson N (1989) Basic algebra. II, 2nd edn. W. H. Freeman and Company, New York Jones [1994] Jones VFR (1994) The Potts model and the symmetric group. In: Subfactors (Kyuzeso, 1993). World Sci. Publ., River Edge, NJ, p 259–267 Kudryavtseva and Mazorchuk [2008] Kudryavtseva G, Mazorchuk V (2008) Schur-Weyl dualities for symmetric inverse semigroups. J Pure Appl Algebra 212(8):1987–1995. 10.1016/j.jpaa.2007.12.004, URL https://doi.org/10.1016/j.jpaa.2007.12.004 Lawson [1998] Lawson MV (1998) Inverse semigroups. World Scientific Publishing Co. Inc., River Edge, NJ, 10.1142/9789812816689, URL http://dx.doi.org/10.1142/9789812816689, the theory of partial symmetries Martin [1994] Martin P (1994) Temperley-Lieb algebras for nonplanar statistical mechanics—the partition algebra construction. J Knot Theory Ramifications 3(1):51–82. 10.1142/S0218216594000071, URL http://dx.doi.org/10.1142/S0218216594000071 Martin [1996] Martin P (1996) The structure of the partition algebras. J Algebra 183(2):319–358. 10.1006/jabr.1996.0223, URL http://dx.doi.org/10.1006/jabr.1996.0223 Martin and Mazorchuk [2014] Martin P, Mazorchuk V (2014) On the representation theory of partial Brauer algebras. Q J Math 65(1):225–247. 10.1093/qmath/has043, URL https://doi.org/10.1093/qmath/has043 Martin [1990] Martin PP (1990) Representations of graph Temperley-Lieb algebras. Publ Res Inst Math Sci 26(3):485–503. 10.2977/prims/1195170958, URL https://doi.org/10.2977/prims/1195170958 Martin [2008] Martin S (2008) Schur algebras and representation theory, Cambridge Tracts in Mathematics, vol 112. Cambridge University Press, Cambridge, reprint of the 1993 original Munn [1955] Munn WD (1955) On semigroup algebras. Proc Cambridge Philos Soc 51:1–15 Munn [1957a] Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Halverson T (2004) Representations of the q𝑞qitalic_q-rook monoid. J Algebra 273(1):227–251. 10.1016/j.jalgebra.2003.11.002, URL http://dx.doi.org/10.1016/j.jalgebra.2003.11.002 Halverson and delMas [2014] Halverson T, delMas E (2014) Representations of the Rook-Brauer algebra. Comm Algebra 42(1):423–443. 10.1080/00927872.2012.716120, URL https://doi.org/10.1080/00927872.2012.716120 Halverson and Ram [2005] Halverson T, Ram A (2005) Partition algebras. European J Combin 26(6):869–921. 10.1016/j.ejc.2004.06.005, URL http://dx.doi.org/10.1016/j.ejc.2004.06.005 Jacobson [1989] Jacobson N (1989) Basic algebra. II, 2nd edn. W. H. Freeman and Company, New York Jones [1994] Jones VFR (1994) The Potts model and the symmetric group. In: Subfactors (Kyuzeso, 1993). World Sci. Publ., River Edge, NJ, p 259–267 Kudryavtseva and Mazorchuk [2008] Kudryavtseva G, Mazorchuk V (2008) Schur-Weyl dualities for symmetric inverse semigroups. J Pure Appl Algebra 212(8):1987–1995. 10.1016/j.jpaa.2007.12.004, URL https://doi.org/10.1016/j.jpaa.2007.12.004 Lawson [1998] Lawson MV (1998) Inverse semigroups. World Scientific Publishing Co. Inc., River Edge, NJ, 10.1142/9789812816689, URL http://dx.doi.org/10.1142/9789812816689, the theory of partial symmetries Martin [1994] Martin P (1994) Temperley-Lieb algebras for nonplanar statistical mechanics—the partition algebra construction. J Knot Theory Ramifications 3(1):51–82. 10.1142/S0218216594000071, URL http://dx.doi.org/10.1142/S0218216594000071 Martin [1996] Martin P (1996) The structure of the partition algebras. J Algebra 183(2):319–358. 10.1006/jabr.1996.0223, URL http://dx.doi.org/10.1006/jabr.1996.0223 Martin and Mazorchuk [2014] Martin P, Mazorchuk V (2014) On the representation theory of partial Brauer algebras. Q J Math 65(1):225–247. 10.1093/qmath/has043, URL https://doi.org/10.1093/qmath/has043 Martin [1990] Martin PP (1990) Representations of graph Temperley-Lieb algebras. Publ Res Inst Math Sci 26(3):485–503. 10.2977/prims/1195170958, URL https://doi.org/10.2977/prims/1195170958 Martin [2008] Martin S (2008) Schur algebras and representation theory, Cambridge Tracts in Mathematics, vol 112. Cambridge University Press, Cambridge, reprint of the 1993 original Munn [1955] Munn WD (1955) On semigroup algebras. Proc Cambridge Philos Soc 51:1–15 Munn [1957a] Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Halverson T, delMas E (2014) Representations of the Rook-Brauer algebra. Comm Algebra 42(1):423–443. 10.1080/00927872.2012.716120, URL https://doi.org/10.1080/00927872.2012.716120 Halverson and Ram [2005] Halverson T, Ram A (2005) Partition algebras. European J Combin 26(6):869–921. 10.1016/j.ejc.2004.06.005, URL http://dx.doi.org/10.1016/j.ejc.2004.06.005 Jacobson [1989] Jacobson N (1989) Basic algebra. II, 2nd edn. W. H. Freeman and Company, New York Jones [1994] Jones VFR (1994) The Potts model and the symmetric group. In: Subfactors (Kyuzeso, 1993). World Sci. Publ., River Edge, NJ, p 259–267 Kudryavtseva and Mazorchuk [2008] Kudryavtseva G, Mazorchuk V (2008) Schur-Weyl dualities for symmetric inverse semigroups. J Pure Appl Algebra 212(8):1987–1995. 10.1016/j.jpaa.2007.12.004, URL https://doi.org/10.1016/j.jpaa.2007.12.004 Lawson [1998] Lawson MV (1998) Inverse semigroups. World Scientific Publishing Co. Inc., River Edge, NJ, 10.1142/9789812816689, URL http://dx.doi.org/10.1142/9789812816689, the theory of partial symmetries Martin [1994] Martin P (1994) Temperley-Lieb algebras for nonplanar statistical mechanics—the partition algebra construction. J Knot Theory Ramifications 3(1):51–82. 10.1142/S0218216594000071, URL http://dx.doi.org/10.1142/S0218216594000071 Martin [1996] Martin P (1996) The structure of the partition algebras. J Algebra 183(2):319–358. 10.1006/jabr.1996.0223, URL http://dx.doi.org/10.1006/jabr.1996.0223 Martin and Mazorchuk [2014] Martin P, Mazorchuk V (2014) On the representation theory of partial Brauer algebras. Q J Math 65(1):225–247. 10.1093/qmath/has043, URL https://doi.org/10.1093/qmath/has043 Martin [1990] Martin PP (1990) Representations of graph Temperley-Lieb algebras. Publ Res Inst Math Sci 26(3):485–503. 10.2977/prims/1195170958, URL https://doi.org/10.2977/prims/1195170958 Martin [2008] Martin S (2008) Schur algebras and representation theory, Cambridge Tracts in Mathematics, vol 112. Cambridge University Press, Cambridge, reprint of the 1993 original Munn [1955] Munn WD (1955) On semigroup algebras. Proc Cambridge Philos Soc 51:1–15 Munn [1957a] Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Halverson T, Ram A (2005) Partition algebras. European J Combin 26(6):869–921. 10.1016/j.ejc.2004.06.005, URL http://dx.doi.org/10.1016/j.ejc.2004.06.005 Jacobson [1989] Jacobson N (1989) Basic algebra. II, 2nd edn. W. H. Freeman and Company, New York Jones [1994] Jones VFR (1994) The Potts model and the symmetric group. In: Subfactors (Kyuzeso, 1993). World Sci. Publ., River Edge, NJ, p 259–267 Kudryavtseva and Mazorchuk [2008] Kudryavtseva G, Mazorchuk V (2008) Schur-Weyl dualities for symmetric inverse semigroups. J Pure Appl Algebra 212(8):1987–1995. 10.1016/j.jpaa.2007.12.004, URL https://doi.org/10.1016/j.jpaa.2007.12.004 Lawson [1998] Lawson MV (1998) Inverse semigroups. World Scientific Publishing Co. Inc., River Edge, NJ, 10.1142/9789812816689, URL http://dx.doi.org/10.1142/9789812816689, the theory of partial symmetries Martin [1994] Martin P (1994) Temperley-Lieb algebras for nonplanar statistical mechanics—the partition algebra construction. J Knot Theory Ramifications 3(1):51–82. 10.1142/S0218216594000071, URL http://dx.doi.org/10.1142/S0218216594000071 Martin [1996] Martin P (1996) The structure of the partition algebras. J Algebra 183(2):319–358. 10.1006/jabr.1996.0223, URL http://dx.doi.org/10.1006/jabr.1996.0223 Martin and Mazorchuk [2014] Martin P, Mazorchuk V (2014) On the representation theory of partial Brauer algebras. Q J Math 65(1):225–247. 10.1093/qmath/has043, URL https://doi.org/10.1093/qmath/has043 Martin [1990] Martin PP (1990) Representations of graph Temperley-Lieb algebras. Publ Res Inst Math Sci 26(3):485–503. 10.2977/prims/1195170958, URL https://doi.org/10.2977/prims/1195170958 Martin [2008] Martin S (2008) Schur algebras and representation theory, Cambridge Tracts in Mathematics, vol 112. Cambridge University Press, Cambridge, reprint of the 1993 original Munn [1955] Munn WD (1955) On semigroup algebras. Proc Cambridge Philos Soc 51:1–15 Munn [1957a] Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Jacobson N (1989) Basic algebra. II, 2nd edn. W. H. Freeman and Company, New York Jones [1994] Jones VFR (1994) The Potts model and the symmetric group. In: Subfactors (Kyuzeso, 1993). World Sci. Publ., River Edge, NJ, p 259–267 Kudryavtseva and Mazorchuk [2008] Kudryavtseva G, Mazorchuk V (2008) Schur-Weyl dualities for symmetric inverse semigroups. J Pure Appl Algebra 212(8):1987–1995. 10.1016/j.jpaa.2007.12.004, URL https://doi.org/10.1016/j.jpaa.2007.12.004 Lawson [1998] Lawson MV (1998) Inverse semigroups. World Scientific Publishing Co. Inc., River Edge, NJ, 10.1142/9789812816689, URL http://dx.doi.org/10.1142/9789812816689, the theory of partial symmetries Martin [1994] Martin P (1994) Temperley-Lieb algebras for nonplanar statistical mechanics—the partition algebra construction. J Knot Theory Ramifications 3(1):51–82. 10.1142/S0218216594000071, URL http://dx.doi.org/10.1142/S0218216594000071 Martin [1996] Martin P (1996) The structure of the partition algebras. J Algebra 183(2):319–358. 10.1006/jabr.1996.0223, URL http://dx.doi.org/10.1006/jabr.1996.0223 Martin and Mazorchuk [2014] Martin P, Mazorchuk V (2014) On the representation theory of partial Brauer algebras. Q J Math 65(1):225–247. 10.1093/qmath/has043, URL https://doi.org/10.1093/qmath/has043 Martin [1990] Martin PP (1990) Representations of graph Temperley-Lieb algebras. Publ Res Inst Math Sci 26(3):485–503. 10.2977/prims/1195170958, URL https://doi.org/10.2977/prims/1195170958 Martin [2008] Martin S (2008) Schur algebras and representation theory, Cambridge Tracts in Mathematics, vol 112. Cambridge University Press, Cambridge, reprint of the 1993 original Munn [1955] Munn WD (1955) On semigroup algebras. Proc Cambridge Philos Soc 51:1–15 Munn [1957a] Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Jones VFR (1994) The Potts model and the symmetric group. In: Subfactors (Kyuzeso, 1993). World Sci. Publ., River Edge, NJ, p 259–267 Kudryavtseva and Mazorchuk [2008] Kudryavtseva G, Mazorchuk V (2008) Schur-Weyl dualities for symmetric inverse semigroups. J Pure Appl Algebra 212(8):1987–1995. 10.1016/j.jpaa.2007.12.004, URL https://doi.org/10.1016/j.jpaa.2007.12.004 Lawson [1998] Lawson MV (1998) Inverse semigroups. World Scientific Publishing Co. Inc., River Edge, NJ, 10.1142/9789812816689, URL http://dx.doi.org/10.1142/9789812816689, the theory of partial symmetries Martin [1994] Martin P (1994) Temperley-Lieb algebras for nonplanar statistical mechanics—the partition algebra construction. J Knot Theory Ramifications 3(1):51–82. 10.1142/S0218216594000071, URL http://dx.doi.org/10.1142/S0218216594000071 Martin [1996] Martin P (1996) The structure of the partition algebras. J Algebra 183(2):319–358. 10.1006/jabr.1996.0223, URL http://dx.doi.org/10.1006/jabr.1996.0223 Martin and Mazorchuk [2014] Martin P, Mazorchuk V (2014) On the representation theory of partial Brauer algebras. Q J Math 65(1):225–247. 10.1093/qmath/has043, URL https://doi.org/10.1093/qmath/has043 Martin [1990] Martin PP (1990) Representations of graph Temperley-Lieb algebras. Publ Res Inst Math Sci 26(3):485–503. 10.2977/prims/1195170958, URL https://doi.org/10.2977/prims/1195170958 Martin [2008] Martin S (2008) Schur algebras and representation theory, Cambridge Tracts in Mathematics, vol 112. Cambridge University Press, Cambridge, reprint of the 1993 original Munn [1955] Munn WD (1955) On semigroup algebras. Proc Cambridge Philos Soc 51:1–15 Munn [1957a] Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Kudryavtseva G, Mazorchuk V (2008) Schur-Weyl dualities for symmetric inverse semigroups. J Pure Appl Algebra 212(8):1987–1995. 10.1016/j.jpaa.2007.12.004, URL https://doi.org/10.1016/j.jpaa.2007.12.004 Lawson [1998] Lawson MV (1998) Inverse semigroups. World Scientific Publishing Co. Inc., River Edge, NJ, 10.1142/9789812816689, URL http://dx.doi.org/10.1142/9789812816689, the theory of partial symmetries Martin [1994] Martin P (1994) Temperley-Lieb algebras for nonplanar statistical mechanics—the partition algebra construction. J Knot Theory Ramifications 3(1):51–82. 10.1142/S0218216594000071, URL http://dx.doi.org/10.1142/S0218216594000071 Martin [1996] Martin P (1996) The structure of the partition algebras. J Algebra 183(2):319–358. 10.1006/jabr.1996.0223, URL http://dx.doi.org/10.1006/jabr.1996.0223 Martin and Mazorchuk [2014] Martin P, Mazorchuk V (2014) On the representation theory of partial Brauer algebras. Q J Math 65(1):225–247. 10.1093/qmath/has043, URL https://doi.org/10.1093/qmath/has043 Martin [1990] Martin PP (1990) Representations of graph Temperley-Lieb algebras. Publ Res Inst Math Sci 26(3):485–503. 10.2977/prims/1195170958, URL https://doi.org/10.2977/prims/1195170958 Martin [2008] Martin S (2008) Schur algebras and representation theory, Cambridge Tracts in Mathematics, vol 112. Cambridge University Press, Cambridge, reprint of the 1993 original Munn [1955] Munn WD (1955) On semigroup algebras. Proc Cambridge Philos Soc 51:1–15 Munn [1957a] Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Lawson MV (1998) Inverse semigroups. World Scientific Publishing Co. Inc., River Edge, NJ, 10.1142/9789812816689, URL http://dx.doi.org/10.1142/9789812816689, the theory of partial symmetries Martin [1994] Martin P (1994) Temperley-Lieb algebras for nonplanar statistical mechanics—the partition algebra construction. J Knot Theory Ramifications 3(1):51–82. 10.1142/S0218216594000071, URL http://dx.doi.org/10.1142/S0218216594000071 Martin [1996] Martin P (1996) The structure of the partition algebras. J Algebra 183(2):319–358. 10.1006/jabr.1996.0223, URL http://dx.doi.org/10.1006/jabr.1996.0223 Martin and Mazorchuk [2014] Martin P, Mazorchuk V (2014) On the representation theory of partial Brauer algebras. Q J Math 65(1):225–247. 10.1093/qmath/has043, URL https://doi.org/10.1093/qmath/has043 Martin [1990] Martin PP (1990) Representations of graph Temperley-Lieb algebras. Publ Res Inst Math Sci 26(3):485–503. 10.2977/prims/1195170958, URL https://doi.org/10.2977/prims/1195170958 Martin [2008] Martin S (2008) Schur algebras and representation theory, Cambridge Tracts in Mathematics, vol 112. Cambridge University Press, Cambridge, reprint of the 1993 original Munn [1955] Munn WD (1955) On semigroup algebras. Proc Cambridge Philos Soc 51:1–15 Munn [1957a] Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Martin P (1994) Temperley-Lieb algebras for nonplanar statistical mechanics—the partition algebra construction. J Knot Theory Ramifications 3(1):51–82. 10.1142/S0218216594000071, URL http://dx.doi.org/10.1142/S0218216594000071 Martin [1996] Martin P (1996) The structure of the partition algebras. J Algebra 183(2):319–358. 10.1006/jabr.1996.0223, URL http://dx.doi.org/10.1006/jabr.1996.0223 Martin and Mazorchuk [2014] Martin P, Mazorchuk V (2014) On the representation theory of partial Brauer algebras. Q J Math 65(1):225–247. 10.1093/qmath/has043, URL https://doi.org/10.1093/qmath/has043 Martin [1990] Martin PP (1990) Representations of graph Temperley-Lieb algebras. Publ Res Inst Math Sci 26(3):485–503. 10.2977/prims/1195170958, URL https://doi.org/10.2977/prims/1195170958 Martin [2008] Martin S (2008) Schur algebras and representation theory, Cambridge Tracts in Mathematics, vol 112. Cambridge University Press, Cambridge, reprint of the 1993 original Munn [1955] Munn WD (1955) On semigroup algebras. Proc Cambridge Philos Soc 51:1–15 Munn [1957a] Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Martin P (1996) The structure of the partition algebras. J Algebra 183(2):319–358. 10.1006/jabr.1996.0223, URL http://dx.doi.org/10.1006/jabr.1996.0223 Martin and Mazorchuk [2014] Martin P, Mazorchuk V (2014) On the representation theory of partial Brauer algebras. Q J Math 65(1):225–247. 10.1093/qmath/has043, URL https://doi.org/10.1093/qmath/has043 Martin [1990] Martin PP (1990) Representations of graph Temperley-Lieb algebras. Publ Res Inst Math Sci 26(3):485–503. 10.2977/prims/1195170958, URL https://doi.org/10.2977/prims/1195170958 Martin [2008] Martin S (2008) Schur algebras and representation theory, Cambridge Tracts in Mathematics, vol 112. Cambridge University Press, Cambridge, reprint of the 1993 original Munn [1955] Munn WD (1955) On semigroup algebras. Proc Cambridge Philos Soc 51:1–15 Munn [1957a] Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Martin P, Mazorchuk V (2014) On the representation theory of partial Brauer algebras. Q J Math 65(1):225–247. 10.1093/qmath/has043, URL https://doi.org/10.1093/qmath/has043 Martin [1990] Martin PP (1990) Representations of graph Temperley-Lieb algebras. Publ Res Inst Math Sci 26(3):485–503. 10.2977/prims/1195170958, URL https://doi.org/10.2977/prims/1195170958 Martin [2008] Martin S (2008) Schur algebras and representation theory, Cambridge Tracts in Mathematics, vol 112. Cambridge University Press, Cambridge, reprint of the 1993 original Munn [1955] Munn WD (1955) On semigroup algebras. Proc Cambridge Philos Soc 51:1–15 Munn [1957a] Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Martin PP (1990) Representations of graph Temperley-Lieb algebras. Publ Res Inst Math Sci 26(3):485–503. 10.2977/prims/1195170958, URL https://doi.org/10.2977/prims/1195170958 Martin [2008] Martin S (2008) Schur algebras and representation theory, Cambridge Tracts in Mathematics, vol 112. Cambridge University Press, Cambridge, reprint of the 1993 original Munn [1955] Munn WD (1955) On semigroup algebras. Proc Cambridge Philos Soc 51:1–15 Munn [1957a] Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Martin S (2008) Schur algebras and representation theory, Cambridge Tracts in Mathematics, vol 112. Cambridge University Press, Cambridge, reprint of the 1993 original Munn [1955] Munn WD (1955) On semigroup algebras. Proc Cambridge Philos Soc 51:1–15 Munn [1957a] Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Munn WD (1955) On semigroup algebras. Proc Cambridge Philos Soc 51:1–15 Munn [1957a] Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J.
- Curtis CW, Reiner I (1981) Methods of representation theory. Vol. I. John Wiley & Sons Inc., New York, with applications to finite groups and orders, Pure and Applied Mathematics, A Wiley-Interscience Publication Dipper et al [2008] Dipper R, Doty S, Hu J (2008) Brauer algebras, symplectic Schur algebras and Schur-Weyl duality. Trans Amer Math Soc 360(1):189–213. 10.1090/S0002-9947-07-04179-7, URL https://doi.org/10.1090/S0002-9947-07-04179-7 Doty and Hu [2009] Doty S, Hu J (2009) Schur-Weyl duality for orthogonal groups. Proc Lond Math Soc (3) 98(3):679–713. 10.1112/plms/pdn044, URL https://doi.org/10.1112/plms/pdn044 Goodman and Wallach [2009] Goodman R, Wallach NR (2009) Symmetry, representations, and invariants, Graduate Texts in Mathematics, vol 255. Springer, Dordrecht, 10.1007/978-0-387-79852-3, URL https://doi.org/10.1007/978-0-387-79852-3 Graham and Lehrer [1996] Graham JJ, Lehrer GI (1996) Cellular algebras. Invent Math 123(1):1–34. 10.1007/BF01232365, URL http://dx.doi.org/10.1007/BF01232365 Green [1980] Green JA (1980) Polynomial representations of GLnsubscriptGL𝑛{\rm GL}_{n}roman_GL start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT, Lecture Notes in Mathematics, vol 830. Springer-Verlag, Berlin Grood [2006] Grood C (2006) The rook partition algebra. J Combin Theory Ser A 113(2):325–351. 10.1016/j.jcta.2005.03.006, URL https://doi.org/10.1016/j.jcta.2005.03.006 Halverson [1996] Halverson T (1996) Characters of the centralizer algebras of mixed tensor representations of Gl(r,ℂ)Gl𝑟ℂ{\rm Gl}(r,\mathbb{C})roman_Gl ( italic_r , blackboard_C ) and the quantum group 𝔘q(ℊl(r,ℂ))subscript𝔘𝑞ℊ𝑙𝑟ℂ\mathfrak{U}_{q}({\mathcal{g}l}(r,\mathbb{C}))fraktur_U start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ( caligraphic_g italic_l ( italic_r , blackboard_C ) ). Pacific J Math 174(2):359–410. URL http://projecteuclid.org/euclid.pjm/1102365176 Halverson [2004] Halverson T (2004) Representations of the q𝑞qitalic_q-rook monoid. J Algebra 273(1):227–251. 10.1016/j.jalgebra.2003.11.002, URL http://dx.doi.org/10.1016/j.jalgebra.2003.11.002 Halverson and delMas [2014] Halverson T, delMas E (2014) Representations of the Rook-Brauer algebra. Comm Algebra 42(1):423–443. 10.1080/00927872.2012.716120, URL https://doi.org/10.1080/00927872.2012.716120 Halverson and Ram [2005] Halverson T, Ram A (2005) Partition algebras. European J Combin 26(6):869–921. 10.1016/j.ejc.2004.06.005, URL http://dx.doi.org/10.1016/j.ejc.2004.06.005 Jacobson [1989] Jacobson N (1989) Basic algebra. II, 2nd edn. W. H. Freeman and Company, New York Jones [1994] Jones VFR (1994) The Potts model and the symmetric group. In: Subfactors (Kyuzeso, 1993). World Sci. Publ., River Edge, NJ, p 259–267 Kudryavtseva and Mazorchuk [2008] Kudryavtseva G, Mazorchuk V (2008) Schur-Weyl dualities for symmetric inverse semigroups. J Pure Appl Algebra 212(8):1987–1995. 10.1016/j.jpaa.2007.12.004, URL https://doi.org/10.1016/j.jpaa.2007.12.004 Lawson [1998] Lawson MV (1998) Inverse semigroups. World Scientific Publishing Co. Inc., River Edge, NJ, 10.1142/9789812816689, URL http://dx.doi.org/10.1142/9789812816689, the theory of partial symmetries Martin [1994] Martin P (1994) Temperley-Lieb algebras for nonplanar statistical mechanics—the partition algebra construction. J Knot Theory Ramifications 3(1):51–82. 10.1142/S0218216594000071, URL http://dx.doi.org/10.1142/S0218216594000071 Martin [1996] Martin P (1996) The structure of the partition algebras. J Algebra 183(2):319–358. 10.1006/jabr.1996.0223, URL http://dx.doi.org/10.1006/jabr.1996.0223 Martin and Mazorchuk [2014] Martin P, Mazorchuk V (2014) On the representation theory of partial Brauer algebras. Q J Math 65(1):225–247. 10.1093/qmath/has043, URL https://doi.org/10.1093/qmath/has043 Martin [1990] Martin PP (1990) Representations of graph Temperley-Lieb algebras. Publ Res Inst Math Sci 26(3):485–503. 10.2977/prims/1195170958, URL https://doi.org/10.2977/prims/1195170958 Martin [2008] Martin S (2008) Schur algebras and representation theory, Cambridge Tracts in Mathematics, vol 112. Cambridge University Press, Cambridge, reprint of the 1993 original Munn [1955] Munn WD (1955) On semigroup algebras. Proc Cambridge Philos Soc 51:1–15 Munn [1957a] Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Dipper R, Doty S, Hu J (2008) Brauer algebras, symplectic Schur algebras and Schur-Weyl duality. Trans Amer Math Soc 360(1):189–213. 10.1090/S0002-9947-07-04179-7, URL https://doi.org/10.1090/S0002-9947-07-04179-7 Doty and Hu [2009] Doty S, Hu J (2009) Schur-Weyl duality for orthogonal groups. Proc Lond Math Soc (3) 98(3):679–713. 10.1112/plms/pdn044, URL https://doi.org/10.1112/plms/pdn044 Goodman and Wallach [2009] Goodman R, Wallach NR (2009) Symmetry, representations, and invariants, Graduate Texts in Mathematics, vol 255. Springer, Dordrecht, 10.1007/978-0-387-79852-3, URL https://doi.org/10.1007/978-0-387-79852-3 Graham and Lehrer [1996] Graham JJ, Lehrer GI (1996) Cellular algebras. Invent Math 123(1):1–34. 10.1007/BF01232365, URL http://dx.doi.org/10.1007/BF01232365 Green [1980] Green JA (1980) Polynomial representations of GLnsubscriptGL𝑛{\rm GL}_{n}roman_GL start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT, Lecture Notes in Mathematics, vol 830. Springer-Verlag, Berlin Grood [2006] Grood C (2006) The rook partition algebra. J Combin Theory Ser A 113(2):325–351. 10.1016/j.jcta.2005.03.006, URL https://doi.org/10.1016/j.jcta.2005.03.006 Halverson [1996] Halverson T (1996) Characters of the centralizer algebras of mixed tensor representations of Gl(r,ℂ)Gl𝑟ℂ{\rm Gl}(r,\mathbb{C})roman_Gl ( italic_r , blackboard_C ) and the quantum group 𝔘q(ℊl(r,ℂ))subscript𝔘𝑞ℊ𝑙𝑟ℂ\mathfrak{U}_{q}({\mathcal{g}l}(r,\mathbb{C}))fraktur_U start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ( caligraphic_g italic_l ( italic_r , blackboard_C ) ). Pacific J Math 174(2):359–410. URL http://projecteuclid.org/euclid.pjm/1102365176 Halverson [2004] Halverson T (2004) Representations of the q𝑞qitalic_q-rook monoid. J Algebra 273(1):227–251. 10.1016/j.jalgebra.2003.11.002, URL http://dx.doi.org/10.1016/j.jalgebra.2003.11.002 Halverson and delMas [2014] Halverson T, delMas E (2014) Representations of the Rook-Brauer algebra. Comm Algebra 42(1):423–443. 10.1080/00927872.2012.716120, URL https://doi.org/10.1080/00927872.2012.716120 Halverson and Ram [2005] Halverson T, Ram A (2005) Partition algebras. European J Combin 26(6):869–921. 10.1016/j.ejc.2004.06.005, URL http://dx.doi.org/10.1016/j.ejc.2004.06.005 Jacobson [1989] Jacobson N (1989) Basic algebra. II, 2nd edn. W. H. Freeman and Company, New York Jones [1994] Jones VFR (1994) The Potts model and the symmetric group. In: Subfactors (Kyuzeso, 1993). World Sci. Publ., River Edge, NJ, p 259–267 Kudryavtseva and Mazorchuk [2008] Kudryavtseva G, Mazorchuk V (2008) Schur-Weyl dualities for symmetric inverse semigroups. J Pure Appl Algebra 212(8):1987–1995. 10.1016/j.jpaa.2007.12.004, URL https://doi.org/10.1016/j.jpaa.2007.12.004 Lawson [1998] Lawson MV (1998) Inverse semigroups. World Scientific Publishing Co. Inc., River Edge, NJ, 10.1142/9789812816689, URL http://dx.doi.org/10.1142/9789812816689, the theory of partial symmetries Martin [1994] Martin P (1994) Temperley-Lieb algebras for nonplanar statistical mechanics—the partition algebra construction. J Knot Theory Ramifications 3(1):51–82. 10.1142/S0218216594000071, URL http://dx.doi.org/10.1142/S0218216594000071 Martin [1996] Martin P (1996) The structure of the partition algebras. J Algebra 183(2):319–358. 10.1006/jabr.1996.0223, URL http://dx.doi.org/10.1006/jabr.1996.0223 Martin and Mazorchuk [2014] Martin P, Mazorchuk V (2014) On the representation theory of partial Brauer algebras. Q J Math 65(1):225–247. 10.1093/qmath/has043, URL https://doi.org/10.1093/qmath/has043 Martin [1990] Martin PP (1990) Representations of graph Temperley-Lieb algebras. Publ Res Inst Math Sci 26(3):485–503. 10.2977/prims/1195170958, URL https://doi.org/10.2977/prims/1195170958 Martin [2008] Martin S (2008) Schur algebras and representation theory, Cambridge Tracts in Mathematics, vol 112. Cambridge University Press, Cambridge, reprint of the 1993 original Munn [1955] Munn WD (1955) On semigroup algebras. Proc Cambridge Philos Soc 51:1–15 Munn [1957a] Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Doty S, Hu J (2009) Schur-Weyl duality for orthogonal groups. Proc Lond Math Soc (3) 98(3):679–713. 10.1112/plms/pdn044, URL https://doi.org/10.1112/plms/pdn044 Goodman and Wallach [2009] Goodman R, Wallach NR (2009) Symmetry, representations, and invariants, Graduate Texts in Mathematics, vol 255. Springer, Dordrecht, 10.1007/978-0-387-79852-3, URL https://doi.org/10.1007/978-0-387-79852-3 Graham and Lehrer [1996] Graham JJ, Lehrer GI (1996) Cellular algebras. Invent Math 123(1):1–34. 10.1007/BF01232365, URL http://dx.doi.org/10.1007/BF01232365 Green [1980] Green JA (1980) Polynomial representations of GLnsubscriptGL𝑛{\rm GL}_{n}roman_GL start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT, Lecture Notes in Mathematics, vol 830. Springer-Verlag, Berlin Grood [2006] Grood C (2006) The rook partition algebra. J Combin Theory Ser A 113(2):325–351. 10.1016/j.jcta.2005.03.006, URL https://doi.org/10.1016/j.jcta.2005.03.006 Halverson [1996] Halverson T (1996) Characters of the centralizer algebras of mixed tensor representations of Gl(r,ℂ)Gl𝑟ℂ{\rm Gl}(r,\mathbb{C})roman_Gl ( italic_r , blackboard_C ) and the quantum group 𝔘q(ℊl(r,ℂ))subscript𝔘𝑞ℊ𝑙𝑟ℂ\mathfrak{U}_{q}({\mathcal{g}l}(r,\mathbb{C}))fraktur_U start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ( caligraphic_g italic_l ( italic_r , blackboard_C ) ). Pacific J Math 174(2):359–410. URL http://projecteuclid.org/euclid.pjm/1102365176 Halverson [2004] Halverson T (2004) Representations of the q𝑞qitalic_q-rook monoid. J Algebra 273(1):227–251. 10.1016/j.jalgebra.2003.11.002, URL http://dx.doi.org/10.1016/j.jalgebra.2003.11.002 Halverson and delMas [2014] Halverson T, delMas E (2014) Representations of the Rook-Brauer algebra. Comm Algebra 42(1):423–443. 10.1080/00927872.2012.716120, URL https://doi.org/10.1080/00927872.2012.716120 Halverson and Ram [2005] Halverson T, Ram A (2005) Partition algebras. European J Combin 26(6):869–921. 10.1016/j.ejc.2004.06.005, URL http://dx.doi.org/10.1016/j.ejc.2004.06.005 Jacobson [1989] Jacobson N (1989) Basic algebra. II, 2nd edn. W. H. Freeman and Company, New York Jones [1994] Jones VFR (1994) The Potts model and the symmetric group. In: Subfactors (Kyuzeso, 1993). World Sci. Publ., River Edge, NJ, p 259–267 Kudryavtseva and Mazorchuk [2008] Kudryavtseva G, Mazorchuk V (2008) Schur-Weyl dualities for symmetric inverse semigroups. J Pure Appl Algebra 212(8):1987–1995. 10.1016/j.jpaa.2007.12.004, URL https://doi.org/10.1016/j.jpaa.2007.12.004 Lawson [1998] Lawson MV (1998) Inverse semigroups. World Scientific Publishing Co. Inc., River Edge, NJ, 10.1142/9789812816689, URL http://dx.doi.org/10.1142/9789812816689, the theory of partial symmetries Martin [1994] Martin P (1994) Temperley-Lieb algebras for nonplanar statistical mechanics—the partition algebra construction. J Knot Theory Ramifications 3(1):51–82. 10.1142/S0218216594000071, URL http://dx.doi.org/10.1142/S0218216594000071 Martin [1996] Martin P (1996) The structure of the partition algebras. J Algebra 183(2):319–358. 10.1006/jabr.1996.0223, URL http://dx.doi.org/10.1006/jabr.1996.0223 Martin and Mazorchuk [2014] Martin P, Mazorchuk V (2014) On the representation theory of partial Brauer algebras. Q J Math 65(1):225–247. 10.1093/qmath/has043, URL https://doi.org/10.1093/qmath/has043 Martin [1990] Martin PP (1990) Representations of graph Temperley-Lieb algebras. Publ Res Inst Math Sci 26(3):485–503. 10.2977/prims/1195170958, URL https://doi.org/10.2977/prims/1195170958 Martin [2008] Martin S (2008) Schur algebras and representation theory, Cambridge Tracts in Mathematics, vol 112. Cambridge University Press, Cambridge, reprint of the 1993 original Munn [1955] Munn WD (1955) On semigroup algebras. Proc Cambridge Philos Soc 51:1–15 Munn [1957a] Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Goodman R, Wallach NR (2009) Symmetry, representations, and invariants, Graduate Texts in Mathematics, vol 255. Springer, Dordrecht, 10.1007/978-0-387-79852-3, URL https://doi.org/10.1007/978-0-387-79852-3 Graham and Lehrer [1996] Graham JJ, Lehrer GI (1996) Cellular algebras. Invent Math 123(1):1–34. 10.1007/BF01232365, URL http://dx.doi.org/10.1007/BF01232365 Green [1980] Green JA (1980) Polynomial representations of GLnsubscriptGL𝑛{\rm GL}_{n}roman_GL start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT, Lecture Notes in Mathematics, vol 830. Springer-Verlag, Berlin Grood [2006] Grood C (2006) The rook partition algebra. J Combin Theory Ser A 113(2):325–351. 10.1016/j.jcta.2005.03.006, URL https://doi.org/10.1016/j.jcta.2005.03.006 Halverson [1996] Halverson T (1996) Characters of the centralizer algebras of mixed tensor representations of Gl(r,ℂ)Gl𝑟ℂ{\rm Gl}(r,\mathbb{C})roman_Gl ( italic_r , blackboard_C ) and the quantum group 𝔘q(ℊl(r,ℂ))subscript𝔘𝑞ℊ𝑙𝑟ℂ\mathfrak{U}_{q}({\mathcal{g}l}(r,\mathbb{C}))fraktur_U start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ( caligraphic_g italic_l ( italic_r , blackboard_C ) ). Pacific J Math 174(2):359–410. URL http://projecteuclid.org/euclid.pjm/1102365176 Halverson [2004] Halverson T (2004) Representations of the q𝑞qitalic_q-rook monoid. J Algebra 273(1):227–251. 10.1016/j.jalgebra.2003.11.002, URL http://dx.doi.org/10.1016/j.jalgebra.2003.11.002 Halverson and delMas [2014] Halverson T, delMas E (2014) Representations of the Rook-Brauer algebra. Comm Algebra 42(1):423–443. 10.1080/00927872.2012.716120, URL https://doi.org/10.1080/00927872.2012.716120 Halverson and Ram [2005] Halverson T, Ram A (2005) Partition algebras. European J Combin 26(6):869–921. 10.1016/j.ejc.2004.06.005, URL http://dx.doi.org/10.1016/j.ejc.2004.06.005 Jacobson [1989] Jacobson N (1989) Basic algebra. II, 2nd edn. W. H. Freeman and Company, New York Jones [1994] Jones VFR (1994) The Potts model and the symmetric group. In: Subfactors (Kyuzeso, 1993). World Sci. Publ., River Edge, NJ, p 259–267 Kudryavtseva and Mazorchuk [2008] Kudryavtseva G, Mazorchuk V (2008) Schur-Weyl dualities for symmetric inverse semigroups. J Pure Appl Algebra 212(8):1987–1995. 10.1016/j.jpaa.2007.12.004, URL https://doi.org/10.1016/j.jpaa.2007.12.004 Lawson [1998] Lawson MV (1998) Inverse semigroups. World Scientific Publishing Co. Inc., River Edge, NJ, 10.1142/9789812816689, URL http://dx.doi.org/10.1142/9789812816689, the theory of partial symmetries Martin [1994] Martin P (1994) Temperley-Lieb algebras for nonplanar statistical mechanics—the partition algebra construction. J Knot Theory Ramifications 3(1):51–82. 10.1142/S0218216594000071, URL http://dx.doi.org/10.1142/S0218216594000071 Martin [1996] Martin P (1996) The structure of the partition algebras. J Algebra 183(2):319–358. 10.1006/jabr.1996.0223, URL http://dx.doi.org/10.1006/jabr.1996.0223 Martin and Mazorchuk [2014] Martin P, Mazorchuk V (2014) On the representation theory of partial Brauer algebras. Q J Math 65(1):225–247. 10.1093/qmath/has043, URL https://doi.org/10.1093/qmath/has043 Martin [1990] Martin PP (1990) Representations of graph Temperley-Lieb algebras. Publ Res Inst Math Sci 26(3):485–503. 10.2977/prims/1195170958, URL https://doi.org/10.2977/prims/1195170958 Martin [2008] Martin S (2008) Schur algebras and representation theory, Cambridge Tracts in Mathematics, vol 112. Cambridge University Press, Cambridge, reprint of the 1993 original Munn [1955] Munn WD (1955) On semigroup algebras. Proc Cambridge Philos Soc 51:1–15 Munn [1957a] Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Graham JJ, Lehrer GI (1996) Cellular algebras. Invent Math 123(1):1–34. 10.1007/BF01232365, URL http://dx.doi.org/10.1007/BF01232365 Green [1980] Green JA (1980) Polynomial representations of GLnsubscriptGL𝑛{\rm GL}_{n}roman_GL start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT, Lecture Notes in Mathematics, vol 830. Springer-Verlag, Berlin Grood [2006] Grood C (2006) The rook partition algebra. J Combin Theory Ser A 113(2):325–351. 10.1016/j.jcta.2005.03.006, URL https://doi.org/10.1016/j.jcta.2005.03.006 Halverson [1996] Halverson T (1996) Characters of the centralizer algebras of mixed tensor representations of Gl(r,ℂ)Gl𝑟ℂ{\rm Gl}(r,\mathbb{C})roman_Gl ( italic_r , blackboard_C ) and the quantum group 𝔘q(ℊl(r,ℂ))subscript𝔘𝑞ℊ𝑙𝑟ℂ\mathfrak{U}_{q}({\mathcal{g}l}(r,\mathbb{C}))fraktur_U start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ( caligraphic_g italic_l ( italic_r , blackboard_C ) ). Pacific J Math 174(2):359–410. URL http://projecteuclid.org/euclid.pjm/1102365176 Halverson [2004] Halverson T (2004) Representations of the q𝑞qitalic_q-rook monoid. J Algebra 273(1):227–251. 10.1016/j.jalgebra.2003.11.002, URL http://dx.doi.org/10.1016/j.jalgebra.2003.11.002 Halverson and delMas [2014] Halverson T, delMas E (2014) Representations of the Rook-Brauer algebra. Comm Algebra 42(1):423–443. 10.1080/00927872.2012.716120, URL https://doi.org/10.1080/00927872.2012.716120 Halverson and Ram [2005] Halverson T, Ram A (2005) Partition algebras. European J Combin 26(6):869–921. 10.1016/j.ejc.2004.06.005, URL http://dx.doi.org/10.1016/j.ejc.2004.06.005 Jacobson [1989] Jacobson N (1989) Basic algebra. II, 2nd edn. W. H. Freeman and Company, New York Jones [1994] Jones VFR (1994) The Potts model and the symmetric group. In: Subfactors (Kyuzeso, 1993). World Sci. Publ., River Edge, NJ, p 259–267 Kudryavtseva and Mazorchuk [2008] Kudryavtseva G, Mazorchuk V (2008) Schur-Weyl dualities for symmetric inverse semigroups. J Pure Appl Algebra 212(8):1987–1995. 10.1016/j.jpaa.2007.12.004, URL https://doi.org/10.1016/j.jpaa.2007.12.004 Lawson [1998] Lawson MV (1998) Inverse semigroups. World Scientific Publishing Co. Inc., River Edge, NJ, 10.1142/9789812816689, URL http://dx.doi.org/10.1142/9789812816689, the theory of partial symmetries Martin [1994] Martin P (1994) Temperley-Lieb algebras for nonplanar statistical mechanics—the partition algebra construction. J Knot Theory Ramifications 3(1):51–82. 10.1142/S0218216594000071, URL http://dx.doi.org/10.1142/S0218216594000071 Martin [1996] Martin P (1996) The structure of the partition algebras. J Algebra 183(2):319–358. 10.1006/jabr.1996.0223, URL http://dx.doi.org/10.1006/jabr.1996.0223 Martin and Mazorchuk [2014] Martin P, Mazorchuk V (2014) On the representation theory of partial Brauer algebras. Q J Math 65(1):225–247. 10.1093/qmath/has043, URL https://doi.org/10.1093/qmath/has043 Martin [1990] Martin PP (1990) Representations of graph Temperley-Lieb algebras. Publ Res Inst Math Sci 26(3):485–503. 10.2977/prims/1195170958, URL https://doi.org/10.2977/prims/1195170958 Martin [2008] Martin S (2008) Schur algebras and representation theory, Cambridge Tracts in Mathematics, vol 112. Cambridge University Press, Cambridge, reprint of the 1993 original Munn [1955] Munn WD (1955) On semigroup algebras. Proc Cambridge Philos Soc 51:1–15 Munn [1957a] Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Green JA (1980) Polynomial representations of GLnsubscriptGL𝑛{\rm GL}_{n}roman_GL start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT, Lecture Notes in Mathematics, vol 830. Springer-Verlag, Berlin Grood [2006] Grood C (2006) The rook partition algebra. J Combin Theory Ser A 113(2):325–351. 10.1016/j.jcta.2005.03.006, URL https://doi.org/10.1016/j.jcta.2005.03.006 Halverson [1996] Halverson T (1996) Characters of the centralizer algebras of mixed tensor representations of Gl(r,ℂ)Gl𝑟ℂ{\rm Gl}(r,\mathbb{C})roman_Gl ( italic_r , blackboard_C ) and the quantum group 𝔘q(ℊl(r,ℂ))subscript𝔘𝑞ℊ𝑙𝑟ℂ\mathfrak{U}_{q}({\mathcal{g}l}(r,\mathbb{C}))fraktur_U start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ( caligraphic_g italic_l ( italic_r , blackboard_C ) ). Pacific J Math 174(2):359–410. URL http://projecteuclid.org/euclid.pjm/1102365176 Halverson [2004] Halverson T (2004) Representations of the q𝑞qitalic_q-rook monoid. J Algebra 273(1):227–251. 10.1016/j.jalgebra.2003.11.002, URL http://dx.doi.org/10.1016/j.jalgebra.2003.11.002 Halverson and delMas [2014] Halverson T, delMas E (2014) Representations of the Rook-Brauer algebra. Comm Algebra 42(1):423–443. 10.1080/00927872.2012.716120, URL https://doi.org/10.1080/00927872.2012.716120 Halverson and Ram [2005] Halverson T, Ram A (2005) Partition algebras. European J Combin 26(6):869–921. 10.1016/j.ejc.2004.06.005, URL http://dx.doi.org/10.1016/j.ejc.2004.06.005 Jacobson [1989] Jacobson N (1989) Basic algebra. II, 2nd edn. W. H. Freeman and Company, New York Jones [1994] Jones VFR (1994) The Potts model and the symmetric group. In: Subfactors (Kyuzeso, 1993). World Sci. Publ., River Edge, NJ, p 259–267 Kudryavtseva and Mazorchuk [2008] Kudryavtseva G, Mazorchuk V (2008) Schur-Weyl dualities for symmetric inverse semigroups. J Pure Appl Algebra 212(8):1987–1995. 10.1016/j.jpaa.2007.12.004, URL https://doi.org/10.1016/j.jpaa.2007.12.004 Lawson [1998] Lawson MV (1998) Inverse semigroups. World Scientific Publishing Co. Inc., River Edge, NJ, 10.1142/9789812816689, URL http://dx.doi.org/10.1142/9789812816689, the theory of partial symmetries Martin [1994] Martin P (1994) Temperley-Lieb algebras for nonplanar statistical mechanics—the partition algebra construction. J Knot Theory Ramifications 3(1):51–82. 10.1142/S0218216594000071, URL http://dx.doi.org/10.1142/S0218216594000071 Martin [1996] Martin P (1996) The structure of the partition algebras. J Algebra 183(2):319–358. 10.1006/jabr.1996.0223, URL http://dx.doi.org/10.1006/jabr.1996.0223 Martin and Mazorchuk [2014] Martin P, Mazorchuk V (2014) On the representation theory of partial Brauer algebras. Q J Math 65(1):225–247. 10.1093/qmath/has043, URL https://doi.org/10.1093/qmath/has043 Martin [1990] Martin PP (1990) Representations of graph Temperley-Lieb algebras. Publ Res Inst Math Sci 26(3):485–503. 10.2977/prims/1195170958, URL https://doi.org/10.2977/prims/1195170958 Martin [2008] Martin S (2008) Schur algebras and representation theory, Cambridge Tracts in Mathematics, vol 112. Cambridge University Press, Cambridge, reprint of the 1993 original Munn [1955] Munn WD (1955) On semigroup algebras. Proc Cambridge Philos Soc 51:1–15 Munn [1957a] Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Grood C (2006) The rook partition algebra. J Combin Theory Ser A 113(2):325–351. 10.1016/j.jcta.2005.03.006, URL https://doi.org/10.1016/j.jcta.2005.03.006 Halverson [1996] Halverson T (1996) Characters of the centralizer algebras of mixed tensor representations of Gl(r,ℂ)Gl𝑟ℂ{\rm Gl}(r,\mathbb{C})roman_Gl ( italic_r , blackboard_C ) and the quantum group 𝔘q(ℊl(r,ℂ))subscript𝔘𝑞ℊ𝑙𝑟ℂ\mathfrak{U}_{q}({\mathcal{g}l}(r,\mathbb{C}))fraktur_U start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ( caligraphic_g italic_l ( italic_r , blackboard_C ) ). Pacific J Math 174(2):359–410. URL http://projecteuclid.org/euclid.pjm/1102365176 Halverson [2004] Halverson T (2004) Representations of the q𝑞qitalic_q-rook monoid. J Algebra 273(1):227–251. 10.1016/j.jalgebra.2003.11.002, URL http://dx.doi.org/10.1016/j.jalgebra.2003.11.002 Halverson and delMas [2014] Halverson T, delMas E (2014) Representations of the Rook-Brauer algebra. Comm Algebra 42(1):423–443. 10.1080/00927872.2012.716120, URL https://doi.org/10.1080/00927872.2012.716120 Halverson and Ram [2005] Halverson T, Ram A (2005) Partition algebras. European J Combin 26(6):869–921. 10.1016/j.ejc.2004.06.005, URL http://dx.doi.org/10.1016/j.ejc.2004.06.005 Jacobson [1989] Jacobson N (1989) Basic algebra. II, 2nd edn. W. H. Freeman and Company, New York Jones [1994] Jones VFR (1994) The Potts model and the symmetric group. In: Subfactors (Kyuzeso, 1993). World Sci. Publ., River Edge, NJ, p 259–267 Kudryavtseva and Mazorchuk [2008] Kudryavtseva G, Mazorchuk V (2008) Schur-Weyl dualities for symmetric inverse semigroups. J Pure Appl Algebra 212(8):1987–1995. 10.1016/j.jpaa.2007.12.004, URL https://doi.org/10.1016/j.jpaa.2007.12.004 Lawson [1998] Lawson MV (1998) Inverse semigroups. World Scientific Publishing Co. Inc., River Edge, NJ, 10.1142/9789812816689, URL http://dx.doi.org/10.1142/9789812816689, the theory of partial symmetries Martin [1994] Martin P (1994) Temperley-Lieb algebras for nonplanar statistical mechanics—the partition algebra construction. J Knot Theory Ramifications 3(1):51–82. 10.1142/S0218216594000071, URL http://dx.doi.org/10.1142/S0218216594000071 Martin [1996] Martin P (1996) The structure of the partition algebras. J Algebra 183(2):319–358. 10.1006/jabr.1996.0223, URL http://dx.doi.org/10.1006/jabr.1996.0223 Martin and Mazorchuk [2014] Martin P, Mazorchuk V (2014) On the representation theory of partial Brauer algebras. Q J Math 65(1):225–247. 10.1093/qmath/has043, URL https://doi.org/10.1093/qmath/has043 Martin [1990] Martin PP (1990) Representations of graph Temperley-Lieb algebras. Publ Res Inst Math Sci 26(3):485–503. 10.2977/prims/1195170958, URL https://doi.org/10.2977/prims/1195170958 Martin [2008] Martin S (2008) Schur algebras and representation theory, Cambridge Tracts in Mathematics, vol 112. Cambridge University Press, Cambridge, reprint of the 1993 original Munn [1955] Munn WD (1955) On semigroup algebras. Proc Cambridge Philos Soc 51:1–15 Munn [1957a] Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Halverson T (1996) Characters of the centralizer algebras of mixed tensor representations of Gl(r,ℂ)Gl𝑟ℂ{\rm Gl}(r,\mathbb{C})roman_Gl ( italic_r , blackboard_C ) and the quantum group 𝔘q(ℊl(r,ℂ))subscript𝔘𝑞ℊ𝑙𝑟ℂ\mathfrak{U}_{q}({\mathcal{g}l}(r,\mathbb{C}))fraktur_U start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ( caligraphic_g italic_l ( italic_r , blackboard_C ) ). Pacific J Math 174(2):359–410. URL http://projecteuclid.org/euclid.pjm/1102365176 Halverson [2004] Halverson T (2004) Representations of the q𝑞qitalic_q-rook monoid. J Algebra 273(1):227–251. 10.1016/j.jalgebra.2003.11.002, URL http://dx.doi.org/10.1016/j.jalgebra.2003.11.002 Halverson and delMas [2014] Halverson T, delMas E (2014) Representations of the Rook-Brauer algebra. Comm Algebra 42(1):423–443. 10.1080/00927872.2012.716120, URL https://doi.org/10.1080/00927872.2012.716120 Halverson and Ram [2005] Halverson T, Ram A (2005) Partition algebras. European J Combin 26(6):869–921. 10.1016/j.ejc.2004.06.005, URL http://dx.doi.org/10.1016/j.ejc.2004.06.005 Jacobson [1989] Jacobson N (1989) Basic algebra. II, 2nd edn. W. H. Freeman and Company, New York Jones [1994] Jones VFR (1994) The Potts model and the symmetric group. In: Subfactors (Kyuzeso, 1993). World Sci. Publ., River Edge, NJ, p 259–267 Kudryavtseva and Mazorchuk [2008] Kudryavtseva G, Mazorchuk V (2008) Schur-Weyl dualities for symmetric inverse semigroups. J Pure Appl Algebra 212(8):1987–1995. 10.1016/j.jpaa.2007.12.004, URL https://doi.org/10.1016/j.jpaa.2007.12.004 Lawson [1998] Lawson MV (1998) Inverse semigroups. World Scientific Publishing Co. Inc., River Edge, NJ, 10.1142/9789812816689, URL http://dx.doi.org/10.1142/9789812816689, the theory of partial symmetries Martin [1994] Martin P (1994) Temperley-Lieb algebras for nonplanar statistical mechanics—the partition algebra construction. J Knot Theory Ramifications 3(1):51–82. 10.1142/S0218216594000071, URL http://dx.doi.org/10.1142/S0218216594000071 Martin [1996] Martin P (1996) The structure of the partition algebras. J Algebra 183(2):319–358. 10.1006/jabr.1996.0223, URL http://dx.doi.org/10.1006/jabr.1996.0223 Martin and Mazorchuk [2014] Martin P, Mazorchuk V (2014) On the representation theory of partial Brauer algebras. Q J Math 65(1):225–247. 10.1093/qmath/has043, URL https://doi.org/10.1093/qmath/has043 Martin [1990] Martin PP (1990) Representations of graph Temperley-Lieb algebras. Publ Res Inst Math Sci 26(3):485–503. 10.2977/prims/1195170958, URL https://doi.org/10.2977/prims/1195170958 Martin [2008] Martin S (2008) Schur algebras and representation theory, Cambridge Tracts in Mathematics, vol 112. Cambridge University Press, Cambridge, reprint of the 1993 original Munn [1955] Munn WD (1955) On semigroup algebras. Proc Cambridge Philos Soc 51:1–15 Munn [1957a] Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Halverson T (2004) Representations of the q𝑞qitalic_q-rook monoid. J Algebra 273(1):227–251. 10.1016/j.jalgebra.2003.11.002, URL http://dx.doi.org/10.1016/j.jalgebra.2003.11.002 Halverson and delMas [2014] Halverson T, delMas E (2014) Representations of the Rook-Brauer algebra. Comm Algebra 42(1):423–443. 10.1080/00927872.2012.716120, URL https://doi.org/10.1080/00927872.2012.716120 Halverson and Ram [2005] Halverson T, Ram A (2005) Partition algebras. European J Combin 26(6):869–921. 10.1016/j.ejc.2004.06.005, URL http://dx.doi.org/10.1016/j.ejc.2004.06.005 Jacobson [1989] Jacobson N (1989) Basic algebra. II, 2nd edn. W. H. Freeman and Company, New York Jones [1994] Jones VFR (1994) The Potts model and the symmetric group. In: Subfactors (Kyuzeso, 1993). World Sci. Publ., River Edge, NJ, p 259–267 Kudryavtseva and Mazorchuk [2008] Kudryavtseva G, Mazorchuk V (2008) Schur-Weyl dualities for symmetric inverse semigroups. J Pure Appl Algebra 212(8):1987–1995. 10.1016/j.jpaa.2007.12.004, URL https://doi.org/10.1016/j.jpaa.2007.12.004 Lawson [1998] Lawson MV (1998) Inverse semigroups. World Scientific Publishing Co. Inc., River Edge, NJ, 10.1142/9789812816689, URL http://dx.doi.org/10.1142/9789812816689, the theory of partial symmetries Martin [1994] Martin P (1994) Temperley-Lieb algebras for nonplanar statistical mechanics—the partition algebra construction. J Knot Theory Ramifications 3(1):51–82. 10.1142/S0218216594000071, URL http://dx.doi.org/10.1142/S0218216594000071 Martin [1996] Martin P (1996) The structure of the partition algebras. J Algebra 183(2):319–358. 10.1006/jabr.1996.0223, URL http://dx.doi.org/10.1006/jabr.1996.0223 Martin and Mazorchuk [2014] Martin P, Mazorchuk V (2014) On the representation theory of partial Brauer algebras. Q J Math 65(1):225–247. 10.1093/qmath/has043, URL https://doi.org/10.1093/qmath/has043 Martin [1990] Martin PP (1990) Representations of graph Temperley-Lieb algebras. Publ Res Inst Math Sci 26(3):485–503. 10.2977/prims/1195170958, URL https://doi.org/10.2977/prims/1195170958 Martin [2008] Martin S (2008) Schur algebras and representation theory, Cambridge Tracts in Mathematics, vol 112. Cambridge University Press, Cambridge, reprint of the 1993 original Munn [1955] Munn WD (1955) On semigroup algebras. Proc Cambridge Philos Soc 51:1–15 Munn [1957a] Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Halverson T, delMas E (2014) Representations of the Rook-Brauer algebra. Comm Algebra 42(1):423–443. 10.1080/00927872.2012.716120, URL https://doi.org/10.1080/00927872.2012.716120 Halverson and Ram [2005] Halverson T, Ram A (2005) Partition algebras. European J Combin 26(6):869–921. 10.1016/j.ejc.2004.06.005, URL http://dx.doi.org/10.1016/j.ejc.2004.06.005 Jacobson [1989] Jacobson N (1989) Basic algebra. II, 2nd edn. W. H. Freeman and Company, New York Jones [1994] Jones VFR (1994) The Potts model and the symmetric group. In: Subfactors (Kyuzeso, 1993). World Sci. Publ., River Edge, NJ, p 259–267 Kudryavtseva and Mazorchuk [2008] Kudryavtseva G, Mazorchuk V (2008) Schur-Weyl dualities for symmetric inverse semigroups. J Pure Appl Algebra 212(8):1987–1995. 10.1016/j.jpaa.2007.12.004, URL https://doi.org/10.1016/j.jpaa.2007.12.004 Lawson [1998] Lawson MV (1998) Inverse semigroups. World Scientific Publishing Co. Inc., River Edge, NJ, 10.1142/9789812816689, URL http://dx.doi.org/10.1142/9789812816689, the theory of partial symmetries Martin [1994] Martin P (1994) Temperley-Lieb algebras for nonplanar statistical mechanics—the partition algebra construction. J Knot Theory Ramifications 3(1):51–82. 10.1142/S0218216594000071, URL http://dx.doi.org/10.1142/S0218216594000071 Martin [1996] Martin P (1996) The structure of the partition algebras. J Algebra 183(2):319–358. 10.1006/jabr.1996.0223, URL http://dx.doi.org/10.1006/jabr.1996.0223 Martin and Mazorchuk [2014] Martin P, Mazorchuk V (2014) On the representation theory of partial Brauer algebras. Q J Math 65(1):225–247. 10.1093/qmath/has043, URL https://doi.org/10.1093/qmath/has043 Martin [1990] Martin PP (1990) Representations of graph Temperley-Lieb algebras. Publ Res Inst Math Sci 26(3):485–503. 10.2977/prims/1195170958, URL https://doi.org/10.2977/prims/1195170958 Martin [2008] Martin S (2008) Schur algebras and representation theory, Cambridge Tracts in Mathematics, vol 112. Cambridge University Press, Cambridge, reprint of the 1993 original Munn [1955] Munn WD (1955) On semigroup algebras. Proc Cambridge Philos Soc 51:1–15 Munn [1957a] Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Halverson T, Ram A (2005) Partition algebras. European J Combin 26(6):869–921. 10.1016/j.ejc.2004.06.005, URL http://dx.doi.org/10.1016/j.ejc.2004.06.005 Jacobson [1989] Jacobson N (1989) Basic algebra. II, 2nd edn. W. H. Freeman and Company, New York Jones [1994] Jones VFR (1994) The Potts model and the symmetric group. In: Subfactors (Kyuzeso, 1993). World Sci. Publ., River Edge, NJ, p 259–267 Kudryavtseva and Mazorchuk [2008] Kudryavtseva G, Mazorchuk V (2008) Schur-Weyl dualities for symmetric inverse semigroups. J Pure Appl Algebra 212(8):1987–1995. 10.1016/j.jpaa.2007.12.004, URL https://doi.org/10.1016/j.jpaa.2007.12.004 Lawson [1998] Lawson MV (1998) Inverse semigroups. World Scientific Publishing Co. Inc., River Edge, NJ, 10.1142/9789812816689, URL http://dx.doi.org/10.1142/9789812816689, the theory of partial symmetries Martin [1994] Martin P (1994) Temperley-Lieb algebras for nonplanar statistical mechanics—the partition algebra construction. J Knot Theory Ramifications 3(1):51–82. 10.1142/S0218216594000071, URL http://dx.doi.org/10.1142/S0218216594000071 Martin [1996] Martin P (1996) The structure of the partition algebras. J Algebra 183(2):319–358. 10.1006/jabr.1996.0223, URL http://dx.doi.org/10.1006/jabr.1996.0223 Martin and Mazorchuk [2014] Martin P, Mazorchuk V (2014) On the representation theory of partial Brauer algebras. Q J Math 65(1):225–247. 10.1093/qmath/has043, URL https://doi.org/10.1093/qmath/has043 Martin [1990] Martin PP (1990) Representations of graph Temperley-Lieb algebras. Publ Res Inst Math Sci 26(3):485–503. 10.2977/prims/1195170958, URL https://doi.org/10.2977/prims/1195170958 Martin [2008] Martin S (2008) Schur algebras and representation theory, Cambridge Tracts in Mathematics, vol 112. Cambridge University Press, Cambridge, reprint of the 1993 original Munn [1955] Munn WD (1955) On semigroup algebras. Proc Cambridge Philos Soc 51:1–15 Munn [1957a] Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Jacobson N (1989) Basic algebra. II, 2nd edn. W. H. Freeman and Company, New York Jones [1994] Jones VFR (1994) The Potts model and the symmetric group. In: Subfactors (Kyuzeso, 1993). World Sci. Publ., River Edge, NJ, p 259–267 Kudryavtseva and Mazorchuk [2008] Kudryavtseva G, Mazorchuk V (2008) Schur-Weyl dualities for symmetric inverse semigroups. J Pure Appl Algebra 212(8):1987–1995. 10.1016/j.jpaa.2007.12.004, URL https://doi.org/10.1016/j.jpaa.2007.12.004 Lawson [1998] Lawson MV (1998) Inverse semigroups. World Scientific Publishing Co. Inc., River Edge, NJ, 10.1142/9789812816689, URL http://dx.doi.org/10.1142/9789812816689, the theory of partial symmetries Martin [1994] Martin P (1994) Temperley-Lieb algebras for nonplanar statistical mechanics—the partition algebra construction. J Knot Theory Ramifications 3(1):51–82. 10.1142/S0218216594000071, URL http://dx.doi.org/10.1142/S0218216594000071 Martin [1996] Martin P (1996) The structure of the partition algebras. J Algebra 183(2):319–358. 10.1006/jabr.1996.0223, URL http://dx.doi.org/10.1006/jabr.1996.0223 Martin and Mazorchuk [2014] Martin P, Mazorchuk V (2014) On the representation theory of partial Brauer algebras. Q J Math 65(1):225–247. 10.1093/qmath/has043, URL https://doi.org/10.1093/qmath/has043 Martin [1990] Martin PP (1990) Representations of graph Temperley-Lieb algebras. Publ Res Inst Math Sci 26(3):485–503. 10.2977/prims/1195170958, URL https://doi.org/10.2977/prims/1195170958 Martin [2008] Martin S (2008) Schur algebras and representation theory, Cambridge Tracts in Mathematics, vol 112. Cambridge University Press, Cambridge, reprint of the 1993 original Munn [1955] Munn WD (1955) On semigroup algebras. Proc Cambridge Philos Soc 51:1–15 Munn [1957a] Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Jones VFR (1994) The Potts model and the symmetric group. In: Subfactors (Kyuzeso, 1993). World Sci. Publ., River Edge, NJ, p 259–267 Kudryavtseva and Mazorchuk [2008] Kudryavtseva G, Mazorchuk V (2008) Schur-Weyl dualities for symmetric inverse semigroups. J Pure Appl Algebra 212(8):1987–1995. 10.1016/j.jpaa.2007.12.004, URL https://doi.org/10.1016/j.jpaa.2007.12.004 Lawson [1998] Lawson MV (1998) Inverse semigroups. World Scientific Publishing Co. Inc., River Edge, NJ, 10.1142/9789812816689, URL http://dx.doi.org/10.1142/9789812816689, the theory of partial symmetries Martin [1994] Martin P (1994) Temperley-Lieb algebras for nonplanar statistical mechanics—the partition algebra construction. J Knot Theory Ramifications 3(1):51–82. 10.1142/S0218216594000071, URL http://dx.doi.org/10.1142/S0218216594000071 Martin [1996] Martin P (1996) The structure of the partition algebras. J Algebra 183(2):319–358. 10.1006/jabr.1996.0223, URL http://dx.doi.org/10.1006/jabr.1996.0223 Martin and Mazorchuk [2014] Martin P, Mazorchuk V (2014) On the representation theory of partial Brauer algebras. Q J Math 65(1):225–247. 10.1093/qmath/has043, URL https://doi.org/10.1093/qmath/has043 Martin [1990] Martin PP (1990) Representations of graph Temperley-Lieb algebras. Publ Res Inst Math Sci 26(3):485–503. 10.2977/prims/1195170958, URL https://doi.org/10.2977/prims/1195170958 Martin [2008] Martin S (2008) Schur algebras and representation theory, Cambridge Tracts in Mathematics, vol 112. Cambridge University Press, Cambridge, reprint of the 1993 original Munn [1955] Munn WD (1955) On semigroup algebras. Proc Cambridge Philos Soc 51:1–15 Munn [1957a] Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Kudryavtseva G, Mazorchuk V (2008) Schur-Weyl dualities for symmetric inverse semigroups. J Pure Appl Algebra 212(8):1987–1995. 10.1016/j.jpaa.2007.12.004, URL https://doi.org/10.1016/j.jpaa.2007.12.004 Lawson [1998] Lawson MV (1998) Inverse semigroups. World Scientific Publishing Co. Inc., River Edge, NJ, 10.1142/9789812816689, URL http://dx.doi.org/10.1142/9789812816689, the theory of partial symmetries Martin [1994] Martin P (1994) Temperley-Lieb algebras for nonplanar statistical mechanics—the partition algebra construction. J Knot Theory Ramifications 3(1):51–82. 10.1142/S0218216594000071, URL http://dx.doi.org/10.1142/S0218216594000071 Martin [1996] Martin P (1996) The structure of the partition algebras. J Algebra 183(2):319–358. 10.1006/jabr.1996.0223, URL http://dx.doi.org/10.1006/jabr.1996.0223 Martin and Mazorchuk [2014] Martin P, Mazorchuk V (2014) On the representation theory of partial Brauer algebras. Q J Math 65(1):225–247. 10.1093/qmath/has043, URL https://doi.org/10.1093/qmath/has043 Martin [1990] Martin PP (1990) Representations of graph Temperley-Lieb algebras. Publ Res Inst Math Sci 26(3):485–503. 10.2977/prims/1195170958, URL https://doi.org/10.2977/prims/1195170958 Martin [2008] Martin S (2008) Schur algebras and representation theory, Cambridge Tracts in Mathematics, vol 112. Cambridge University Press, Cambridge, reprint of the 1993 original Munn [1955] Munn WD (1955) On semigroup algebras. Proc Cambridge Philos Soc 51:1–15 Munn [1957a] Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Lawson MV (1998) Inverse semigroups. World Scientific Publishing Co. Inc., River Edge, NJ, 10.1142/9789812816689, URL http://dx.doi.org/10.1142/9789812816689, the theory of partial symmetries Martin [1994] Martin P (1994) Temperley-Lieb algebras for nonplanar statistical mechanics—the partition algebra construction. J Knot Theory Ramifications 3(1):51–82. 10.1142/S0218216594000071, URL http://dx.doi.org/10.1142/S0218216594000071 Martin [1996] Martin P (1996) The structure of the partition algebras. J Algebra 183(2):319–358. 10.1006/jabr.1996.0223, URL http://dx.doi.org/10.1006/jabr.1996.0223 Martin and Mazorchuk [2014] Martin P, Mazorchuk V (2014) On the representation theory of partial Brauer algebras. Q J Math 65(1):225–247. 10.1093/qmath/has043, URL https://doi.org/10.1093/qmath/has043 Martin [1990] Martin PP (1990) Representations of graph Temperley-Lieb algebras. Publ Res Inst Math Sci 26(3):485–503. 10.2977/prims/1195170958, URL https://doi.org/10.2977/prims/1195170958 Martin [2008] Martin S (2008) Schur algebras and representation theory, Cambridge Tracts in Mathematics, vol 112. Cambridge University Press, Cambridge, reprint of the 1993 original Munn [1955] Munn WD (1955) On semigroup algebras. Proc Cambridge Philos Soc 51:1–15 Munn [1957a] Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Martin P (1994) Temperley-Lieb algebras for nonplanar statistical mechanics—the partition algebra construction. J Knot Theory Ramifications 3(1):51–82. 10.1142/S0218216594000071, URL http://dx.doi.org/10.1142/S0218216594000071 Martin [1996] Martin P (1996) The structure of the partition algebras. J Algebra 183(2):319–358. 10.1006/jabr.1996.0223, URL http://dx.doi.org/10.1006/jabr.1996.0223 Martin and Mazorchuk [2014] Martin P, Mazorchuk V (2014) On the representation theory of partial Brauer algebras. Q J Math 65(1):225–247. 10.1093/qmath/has043, URL https://doi.org/10.1093/qmath/has043 Martin [1990] Martin PP (1990) Representations of graph Temperley-Lieb algebras. Publ Res Inst Math Sci 26(3):485–503. 10.2977/prims/1195170958, URL https://doi.org/10.2977/prims/1195170958 Martin [2008] Martin S (2008) Schur algebras and representation theory, Cambridge Tracts in Mathematics, vol 112. Cambridge University Press, Cambridge, reprint of the 1993 original Munn [1955] Munn WD (1955) On semigroup algebras. Proc Cambridge Philos Soc 51:1–15 Munn [1957a] Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Martin P (1996) The structure of the partition algebras. J Algebra 183(2):319–358. 10.1006/jabr.1996.0223, URL http://dx.doi.org/10.1006/jabr.1996.0223 Martin and Mazorchuk [2014] Martin P, Mazorchuk V (2014) On the representation theory of partial Brauer algebras. Q J Math 65(1):225–247. 10.1093/qmath/has043, URL https://doi.org/10.1093/qmath/has043 Martin [1990] Martin PP (1990) Representations of graph Temperley-Lieb algebras. Publ Res Inst Math Sci 26(3):485–503. 10.2977/prims/1195170958, URL https://doi.org/10.2977/prims/1195170958 Martin [2008] Martin S (2008) Schur algebras and representation theory, Cambridge Tracts in Mathematics, vol 112. Cambridge University Press, Cambridge, reprint of the 1993 original Munn [1955] Munn WD (1955) On semigroup algebras. Proc Cambridge Philos Soc 51:1–15 Munn [1957a] Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Martin P, Mazorchuk V (2014) On the representation theory of partial Brauer algebras. Q J Math 65(1):225–247. 10.1093/qmath/has043, URL https://doi.org/10.1093/qmath/has043 Martin [1990] Martin PP (1990) Representations of graph Temperley-Lieb algebras. Publ Res Inst Math Sci 26(3):485–503. 10.2977/prims/1195170958, URL https://doi.org/10.2977/prims/1195170958 Martin [2008] Martin S (2008) Schur algebras and representation theory, Cambridge Tracts in Mathematics, vol 112. Cambridge University Press, Cambridge, reprint of the 1993 original Munn [1955] Munn WD (1955) On semigroup algebras. Proc Cambridge Philos Soc 51:1–15 Munn [1957a] Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Martin PP (1990) Representations of graph Temperley-Lieb algebras. Publ Res Inst Math Sci 26(3):485–503. 10.2977/prims/1195170958, URL https://doi.org/10.2977/prims/1195170958 Martin [2008] Martin S (2008) Schur algebras and representation theory, Cambridge Tracts in Mathematics, vol 112. Cambridge University Press, Cambridge, reprint of the 1993 original Munn [1955] Munn WD (1955) On semigroup algebras. Proc Cambridge Philos Soc 51:1–15 Munn [1957a] Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Martin S (2008) Schur algebras and representation theory, Cambridge Tracts in Mathematics, vol 112. Cambridge University Press, Cambridge, reprint of the 1993 original Munn [1955] Munn WD (1955) On semigroup algebras. Proc Cambridge Philos Soc 51:1–15 Munn [1957a] Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Munn WD (1955) On semigroup algebras. Proc Cambridge Philos Soc 51:1–15 Munn [1957a] Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J.
- Dipper R, Doty S, Hu J (2008) Brauer algebras, symplectic Schur algebras and Schur-Weyl duality. Trans Amer Math Soc 360(1):189–213. 10.1090/S0002-9947-07-04179-7, URL https://doi.org/10.1090/S0002-9947-07-04179-7 Doty and Hu [2009] Doty S, Hu J (2009) Schur-Weyl duality for orthogonal groups. Proc Lond Math Soc (3) 98(3):679–713. 10.1112/plms/pdn044, URL https://doi.org/10.1112/plms/pdn044 Goodman and Wallach [2009] Goodman R, Wallach NR (2009) Symmetry, representations, and invariants, Graduate Texts in Mathematics, vol 255. Springer, Dordrecht, 10.1007/978-0-387-79852-3, URL https://doi.org/10.1007/978-0-387-79852-3 Graham and Lehrer [1996] Graham JJ, Lehrer GI (1996) Cellular algebras. Invent Math 123(1):1–34. 10.1007/BF01232365, URL http://dx.doi.org/10.1007/BF01232365 Green [1980] Green JA (1980) Polynomial representations of GLnsubscriptGL𝑛{\rm GL}_{n}roman_GL start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT, Lecture Notes in Mathematics, vol 830. Springer-Verlag, Berlin Grood [2006] Grood C (2006) The rook partition algebra. J Combin Theory Ser A 113(2):325–351. 10.1016/j.jcta.2005.03.006, URL https://doi.org/10.1016/j.jcta.2005.03.006 Halverson [1996] Halverson T (1996) Characters of the centralizer algebras of mixed tensor representations of Gl(r,ℂ)Gl𝑟ℂ{\rm Gl}(r,\mathbb{C})roman_Gl ( italic_r , blackboard_C ) and the quantum group 𝔘q(ℊl(r,ℂ))subscript𝔘𝑞ℊ𝑙𝑟ℂ\mathfrak{U}_{q}({\mathcal{g}l}(r,\mathbb{C}))fraktur_U start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ( caligraphic_g italic_l ( italic_r , blackboard_C ) ). Pacific J Math 174(2):359–410. URL http://projecteuclid.org/euclid.pjm/1102365176 Halverson [2004] Halverson T (2004) Representations of the q𝑞qitalic_q-rook monoid. J Algebra 273(1):227–251. 10.1016/j.jalgebra.2003.11.002, URL http://dx.doi.org/10.1016/j.jalgebra.2003.11.002 Halverson and delMas [2014] Halverson T, delMas E (2014) Representations of the Rook-Brauer algebra. Comm Algebra 42(1):423–443. 10.1080/00927872.2012.716120, URL https://doi.org/10.1080/00927872.2012.716120 Halverson and Ram [2005] Halverson T, Ram A (2005) Partition algebras. European J Combin 26(6):869–921. 10.1016/j.ejc.2004.06.005, URL http://dx.doi.org/10.1016/j.ejc.2004.06.005 Jacobson [1989] Jacobson N (1989) Basic algebra. II, 2nd edn. W. H. Freeman and Company, New York Jones [1994] Jones VFR (1994) The Potts model and the symmetric group. In: Subfactors (Kyuzeso, 1993). World Sci. Publ., River Edge, NJ, p 259–267 Kudryavtseva and Mazorchuk [2008] Kudryavtseva G, Mazorchuk V (2008) Schur-Weyl dualities for symmetric inverse semigroups. J Pure Appl Algebra 212(8):1987–1995. 10.1016/j.jpaa.2007.12.004, URL https://doi.org/10.1016/j.jpaa.2007.12.004 Lawson [1998] Lawson MV (1998) Inverse semigroups. World Scientific Publishing Co. Inc., River Edge, NJ, 10.1142/9789812816689, URL http://dx.doi.org/10.1142/9789812816689, the theory of partial symmetries Martin [1994] Martin P (1994) Temperley-Lieb algebras for nonplanar statistical mechanics—the partition algebra construction. J Knot Theory Ramifications 3(1):51–82. 10.1142/S0218216594000071, URL http://dx.doi.org/10.1142/S0218216594000071 Martin [1996] Martin P (1996) The structure of the partition algebras. J Algebra 183(2):319–358. 10.1006/jabr.1996.0223, URL http://dx.doi.org/10.1006/jabr.1996.0223 Martin and Mazorchuk [2014] Martin P, Mazorchuk V (2014) On the representation theory of partial Brauer algebras. Q J Math 65(1):225–247. 10.1093/qmath/has043, URL https://doi.org/10.1093/qmath/has043 Martin [1990] Martin PP (1990) Representations of graph Temperley-Lieb algebras. Publ Res Inst Math Sci 26(3):485–503. 10.2977/prims/1195170958, URL https://doi.org/10.2977/prims/1195170958 Martin [2008] Martin S (2008) Schur algebras and representation theory, Cambridge Tracts in Mathematics, vol 112. Cambridge University Press, Cambridge, reprint of the 1993 original Munn [1955] Munn WD (1955) On semigroup algebras. Proc Cambridge Philos Soc 51:1–15 Munn [1957a] Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Doty S, Hu J (2009) Schur-Weyl duality for orthogonal groups. Proc Lond Math Soc (3) 98(3):679–713. 10.1112/plms/pdn044, URL https://doi.org/10.1112/plms/pdn044 Goodman and Wallach [2009] Goodman R, Wallach NR (2009) Symmetry, representations, and invariants, Graduate Texts in Mathematics, vol 255. Springer, Dordrecht, 10.1007/978-0-387-79852-3, URL https://doi.org/10.1007/978-0-387-79852-3 Graham and Lehrer [1996] Graham JJ, Lehrer GI (1996) Cellular algebras. Invent Math 123(1):1–34. 10.1007/BF01232365, URL http://dx.doi.org/10.1007/BF01232365 Green [1980] Green JA (1980) Polynomial representations of GLnsubscriptGL𝑛{\rm GL}_{n}roman_GL start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT, Lecture Notes in Mathematics, vol 830. Springer-Verlag, Berlin Grood [2006] Grood C (2006) The rook partition algebra. J Combin Theory Ser A 113(2):325–351. 10.1016/j.jcta.2005.03.006, URL https://doi.org/10.1016/j.jcta.2005.03.006 Halverson [1996] Halverson T (1996) Characters of the centralizer algebras of mixed tensor representations of Gl(r,ℂ)Gl𝑟ℂ{\rm Gl}(r,\mathbb{C})roman_Gl ( italic_r , blackboard_C ) and the quantum group 𝔘q(ℊl(r,ℂ))subscript𝔘𝑞ℊ𝑙𝑟ℂ\mathfrak{U}_{q}({\mathcal{g}l}(r,\mathbb{C}))fraktur_U start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ( caligraphic_g italic_l ( italic_r , blackboard_C ) ). Pacific J Math 174(2):359–410. URL http://projecteuclid.org/euclid.pjm/1102365176 Halverson [2004] Halverson T (2004) Representations of the q𝑞qitalic_q-rook monoid. J Algebra 273(1):227–251. 10.1016/j.jalgebra.2003.11.002, URL http://dx.doi.org/10.1016/j.jalgebra.2003.11.002 Halverson and delMas [2014] Halverson T, delMas E (2014) Representations of the Rook-Brauer algebra. Comm Algebra 42(1):423–443. 10.1080/00927872.2012.716120, URL https://doi.org/10.1080/00927872.2012.716120 Halverson and Ram [2005] Halverson T, Ram A (2005) Partition algebras. European J Combin 26(6):869–921. 10.1016/j.ejc.2004.06.005, URL http://dx.doi.org/10.1016/j.ejc.2004.06.005 Jacobson [1989] Jacobson N (1989) Basic algebra. II, 2nd edn. W. H. Freeman and Company, New York Jones [1994] Jones VFR (1994) The Potts model and the symmetric group. In: Subfactors (Kyuzeso, 1993). World Sci. Publ., River Edge, NJ, p 259–267 Kudryavtseva and Mazorchuk [2008] Kudryavtseva G, Mazorchuk V (2008) Schur-Weyl dualities for symmetric inverse semigroups. J Pure Appl Algebra 212(8):1987–1995. 10.1016/j.jpaa.2007.12.004, URL https://doi.org/10.1016/j.jpaa.2007.12.004 Lawson [1998] Lawson MV (1998) Inverse semigroups. World Scientific Publishing Co. Inc., River Edge, NJ, 10.1142/9789812816689, URL http://dx.doi.org/10.1142/9789812816689, the theory of partial symmetries Martin [1994] Martin P (1994) Temperley-Lieb algebras for nonplanar statistical mechanics—the partition algebra construction. J Knot Theory Ramifications 3(1):51–82. 10.1142/S0218216594000071, URL http://dx.doi.org/10.1142/S0218216594000071 Martin [1996] Martin P (1996) The structure of the partition algebras. J Algebra 183(2):319–358. 10.1006/jabr.1996.0223, URL http://dx.doi.org/10.1006/jabr.1996.0223 Martin and Mazorchuk [2014] Martin P, Mazorchuk V (2014) On the representation theory of partial Brauer algebras. Q J Math 65(1):225–247. 10.1093/qmath/has043, URL https://doi.org/10.1093/qmath/has043 Martin [1990] Martin PP (1990) Representations of graph Temperley-Lieb algebras. Publ Res Inst Math Sci 26(3):485–503. 10.2977/prims/1195170958, URL https://doi.org/10.2977/prims/1195170958 Martin [2008] Martin S (2008) Schur algebras and representation theory, Cambridge Tracts in Mathematics, vol 112. Cambridge University Press, Cambridge, reprint of the 1993 original Munn [1955] Munn WD (1955) On semigroup algebras. Proc Cambridge Philos Soc 51:1–15 Munn [1957a] Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Goodman R, Wallach NR (2009) Symmetry, representations, and invariants, Graduate Texts in Mathematics, vol 255. Springer, Dordrecht, 10.1007/978-0-387-79852-3, URL https://doi.org/10.1007/978-0-387-79852-3 Graham and Lehrer [1996] Graham JJ, Lehrer GI (1996) Cellular algebras. Invent Math 123(1):1–34. 10.1007/BF01232365, URL http://dx.doi.org/10.1007/BF01232365 Green [1980] Green JA (1980) Polynomial representations of GLnsubscriptGL𝑛{\rm GL}_{n}roman_GL start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT, Lecture Notes in Mathematics, vol 830. Springer-Verlag, Berlin Grood [2006] Grood C (2006) The rook partition algebra. J Combin Theory Ser A 113(2):325–351. 10.1016/j.jcta.2005.03.006, URL https://doi.org/10.1016/j.jcta.2005.03.006 Halverson [1996] Halverson T (1996) Characters of the centralizer algebras of mixed tensor representations of Gl(r,ℂ)Gl𝑟ℂ{\rm Gl}(r,\mathbb{C})roman_Gl ( italic_r , blackboard_C ) and the quantum group 𝔘q(ℊl(r,ℂ))subscript𝔘𝑞ℊ𝑙𝑟ℂ\mathfrak{U}_{q}({\mathcal{g}l}(r,\mathbb{C}))fraktur_U start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ( caligraphic_g italic_l ( italic_r , blackboard_C ) ). Pacific J Math 174(2):359–410. URL http://projecteuclid.org/euclid.pjm/1102365176 Halverson [2004] Halverson T (2004) Representations of the q𝑞qitalic_q-rook monoid. J Algebra 273(1):227–251. 10.1016/j.jalgebra.2003.11.002, URL http://dx.doi.org/10.1016/j.jalgebra.2003.11.002 Halverson and delMas [2014] Halverson T, delMas E (2014) Representations of the Rook-Brauer algebra. Comm Algebra 42(1):423–443. 10.1080/00927872.2012.716120, URL https://doi.org/10.1080/00927872.2012.716120 Halverson and Ram [2005] Halverson T, Ram A (2005) Partition algebras. European J Combin 26(6):869–921. 10.1016/j.ejc.2004.06.005, URL http://dx.doi.org/10.1016/j.ejc.2004.06.005 Jacobson [1989] Jacobson N (1989) Basic algebra. II, 2nd edn. W. H. Freeman and Company, New York Jones [1994] Jones VFR (1994) The Potts model and the symmetric group. In: Subfactors (Kyuzeso, 1993). World Sci. Publ., River Edge, NJ, p 259–267 Kudryavtseva and Mazorchuk [2008] Kudryavtseva G, Mazorchuk V (2008) Schur-Weyl dualities for symmetric inverse semigroups. J Pure Appl Algebra 212(8):1987–1995. 10.1016/j.jpaa.2007.12.004, URL https://doi.org/10.1016/j.jpaa.2007.12.004 Lawson [1998] Lawson MV (1998) Inverse semigroups. World Scientific Publishing Co. Inc., River Edge, NJ, 10.1142/9789812816689, URL http://dx.doi.org/10.1142/9789812816689, the theory of partial symmetries Martin [1994] Martin P (1994) Temperley-Lieb algebras for nonplanar statistical mechanics—the partition algebra construction. J Knot Theory Ramifications 3(1):51–82. 10.1142/S0218216594000071, URL http://dx.doi.org/10.1142/S0218216594000071 Martin [1996] Martin P (1996) The structure of the partition algebras. J Algebra 183(2):319–358. 10.1006/jabr.1996.0223, URL http://dx.doi.org/10.1006/jabr.1996.0223 Martin and Mazorchuk [2014] Martin P, Mazorchuk V (2014) On the representation theory of partial Brauer algebras. Q J Math 65(1):225–247. 10.1093/qmath/has043, URL https://doi.org/10.1093/qmath/has043 Martin [1990] Martin PP (1990) Representations of graph Temperley-Lieb algebras. Publ Res Inst Math Sci 26(3):485–503. 10.2977/prims/1195170958, URL https://doi.org/10.2977/prims/1195170958 Martin [2008] Martin S (2008) Schur algebras and representation theory, Cambridge Tracts in Mathematics, vol 112. Cambridge University Press, Cambridge, reprint of the 1993 original Munn [1955] Munn WD (1955) On semigroup algebras. Proc Cambridge Philos Soc 51:1–15 Munn [1957a] Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Graham JJ, Lehrer GI (1996) Cellular algebras. Invent Math 123(1):1–34. 10.1007/BF01232365, URL http://dx.doi.org/10.1007/BF01232365 Green [1980] Green JA (1980) Polynomial representations of GLnsubscriptGL𝑛{\rm GL}_{n}roman_GL start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT, Lecture Notes in Mathematics, vol 830. Springer-Verlag, Berlin Grood [2006] Grood C (2006) The rook partition algebra. J Combin Theory Ser A 113(2):325–351. 10.1016/j.jcta.2005.03.006, URL https://doi.org/10.1016/j.jcta.2005.03.006 Halverson [1996] Halverson T (1996) Characters of the centralizer algebras of mixed tensor representations of Gl(r,ℂ)Gl𝑟ℂ{\rm Gl}(r,\mathbb{C})roman_Gl ( italic_r , blackboard_C ) and the quantum group 𝔘q(ℊl(r,ℂ))subscript𝔘𝑞ℊ𝑙𝑟ℂ\mathfrak{U}_{q}({\mathcal{g}l}(r,\mathbb{C}))fraktur_U start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ( caligraphic_g italic_l ( italic_r , blackboard_C ) ). Pacific J Math 174(2):359–410. URL http://projecteuclid.org/euclid.pjm/1102365176 Halverson [2004] Halverson T (2004) Representations of the q𝑞qitalic_q-rook monoid. J Algebra 273(1):227–251. 10.1016/j.jalgebra.2003.11.002, URL http://dx.doi.org/10.1016/j.jalgebra.2003.11.002 Halverson and delMas [2014] Halverson T, delMas E (2014) Representations of the Rook-Brauer algebra. Comm Algebra 42(1):423–443. 10.1080/00927872.2012.716120, URL https://doi.org/10.1080/00927872.2012.716120 Halverson and Ram [2005] Halverson T, Ram A (2005) Partition algebras. European J Combin 26(6):869–921. 10.1016/j.ejc.2004.06.005, URL http://dx.doi.org/10.1016/j.ejc.2004.06.005 Jacobson [1989] Jacobson N (1989) Basic algebra. II, 2nd edn. W. H. Freeman and Company, New York Jones [1994] Jones VFR (1994) The Potts model and the symmetric group. In: Subfactors (Kyuzeso, 1993). World Sci. Publ., River Edge, NJ, p 259–267 Kudryavtseva and Mazorchuk [2008] Kudryavtseva G, Mazorchuk V (2008) Schur-Weyl dualities for symmetric inverse semigroups. J Pure Appl Algebra 212(8):1987–1995. 10.1016/j.jpaa.2007.12.004, URL https://doi.org/10.1016/j.jpaa.2007.12.004 Lawson [1998] Lawson MV (1998) Inverse semigroups. World Scientific Publishing Co. Inc., River Edge, NJ, 10.1142/9789812816689, URL http://dx.doi.org/10.1142/9789812816689, the theory of partial symmetries Martin [1994] Martin P (1994) Temperley-Lieb algebras for nonplanar statistical mechanics—the partition algebra construction. J Knot Theory Ramifications 3(1):51–82. 10.1142/S0218216594000071, URL http://dx.doi.org/10.1142/S0218216594000071 Martin [1996] Martin P (1996) The structure of the partition algebras. J Algebra 183(2):319–358. 10.1006/jabr.1996.0223, URL http://dx.doi.org/10.1006/jabr.1996.0223 Martin and Mazorchuk [2014] Martin P, Mazorchuk V (2014) On the representation theory of partial Brauer algebras. Q J Math 65(1):225–247. 10.1093/qmath/has043, URL https://doi.org/10.1093/qmath/has043 Martin [1990] Martin PP (1990) Representations of graph Temperley-Lieb algebras. Publ Res Inst Math Sci 26(3):485–503. 10.2977/prims/1195170958, URL https://doi.org/10.2977/prims/1195170958 Martin [2008] Martin S (2008) Schur algebras and representation theory, Cambridge Tracts in Mathematics, vol 112. Cambridge University Press, Cambridge, reprint of the 1993 original Munn [1955] Munn WD (1955) On semigroup algebras. Proc Cambridge Philos Soc 51:1–15 Munn [1957a] Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Green JA (1980) Polynomial representations of GLnsubscriptGL𝑛{\rm GL}_{n}roman_GL start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT, Lecture Notes in Mathematics, vol 830. Springer-Verlag, Berlin Grood [2006] Grood C (2006) The rook partition algebra. J Combin Theory Ser A 113(2):325–351. 10.1016/j.jcta.2005.03.006, URL https://doi.org/10.1016/j.jcta.2005.03.006 Halverson [1996] Halverson T (1996) Characters of the centralizer algebras of mixed tensor representations of Gl(r,ℂ)Gl𝑟ℂ{\rm Gl}(r,\mathbb{C})roman_Gl ( italic_r , blackboard_C ) and the quantum group 𝔘q(ℊl(r,ℂ))subscript𝔘𝑞ℊ𝑙𝑟ℂ\mathfrak{U}_{q}({\mathcal{g}l}(r,\mathbb{C}))fraktur_U start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ( caligraphic_g italic_l ( italic_r , blackboard_C ) ). Pacific J Math 174(2):359–410. URL http://projecteuclid.org/euclid.pjm/1102365176 Halverson [2004] Halverson T (2004) Representations of the q𝑞qitalic_q-rook monoid. J Algebra 273(1):227–251. 10.1016/j.jalgebra.2003.11.002, URL http://dx.doi.org/10.1016/j.jalgebra.2003.11.002 Halverson and delMas [2014] Halverson T, delMas E (2014) Representations of the Rook-Brauer algebra. Comm Algebra 42(1):423–443. 10.1080/00927872.2012.716120, URL https://doi.org/10.1080/00927872.2012.716120 Halverson and Ram [2005] Halverson T, Ram A (2005) Partition algebras. European J Combin 26(6):869–921. 10.1016/j.ejc.2004.06.005, URL http://dx.doi.org/10.1016/j.ejc.2004.06.005 Jacobson [1989] Jacobson N (1989) Basic algebra. II, 2nd edn. W. H. Freeman and Company, New York Jones [1994] Jones VFR (1994) The Potts model and the symmetric group. In: Subfactors (Kyuzeso, 1993). World Sci. Publ., River Edge, NJ, p 259–267 Kudryavtseva and Mazorchuk [2008] Kudryavtseva G, Mazorchuk V (2008) Schur-Weyl dualities for symmetric inverse semigroups. J Pure Appl Algebra 212(8):1987–1995. 10.1016/j.jpaa.2007.12.004, URL https://doi.org/10.1016/j.jpaa.2007.12.004 Lawson [1998] Lawson MV (1998) Inverse semigroups. World Scientific Publishing Co. Inc., River Edge, NJ, 10.1142/9789812816689, URL http://dx.doi.org/10.1142/9789812816689, the theory of partial symmetries Martin [1994] Martin P (1994) Temperley-Lieb algebras for nonplanar statistical mechanics—the partition algebra construction. J Knot Theory Ramifications 3(1):51–82. 10.1142/S0218216594000071, URL http://dx.doi.org/10.1142/S0218216594000071 Martin [1996] Martin P (1996) The structure of the partition algebras. J Algebra 183(2):319–358. 10.1006/jabr.1996.0223, URL http://dx.doi.org/10.1006/jabr.1996.0223 Martin and Mazorchuk [2014] Martin P, Mazorchuk V (2014) On the representation theory of partial Brauer algebras. Q J Math 65(1):225–247. 10.1093/qmath/has043, URL https://doi.org/10.1093/qmath/has043 Martin [1990] Martin PP (1990) Representations of graph Temperley-Lieb algebras. Publ Res Inst Math Sci 26(3):485–503. 10.2977/prims/1195170958, URL https://doi.org/10.2977/prims/1195170958 Martin [2008] Martin S (2008) Schur algebras and representation theory, Cambridge Tracts in Mathematics, vol 112. Cambridge University Press, Cambridge, reprint of the 1993 original Munn [1955] Munn WD (1955) On semigroup algebras. Proc Cambridge Philos Soc 51:1–15 Munn [1957a] Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Grood C (2006) The rook partition algebra. J Combin Theory Ser A 113(2):325–351. 10.1016/j.jcta.2005.03.006, URL https://doi.org/10.1016/j.jcta.2005.03.006 Halverson [1996] Halverson T (1996) Characters of the centralizer algebras of mixed tensor representations of Gl(r,ℂ)Gl𝑟ℂ{\rm Gl}(r,\mathbb{C})roman_Gl ( italic_r , blackboard_C ) and the quantum group 𝔘q(ℊl(r,ℂ))subscript𝔘𝑞ℊ𝑙𝑟ℂ\mathfrak{U}_{q}({\mathcal{g}l}(r,\mathbb{C}))fraktur_U start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ( caligraphic_g italic_l ( italic_r , blackboard_C ) ). Pacific J Math 174(2):359–410. URL http://projecteuclid.org/euclid.pjm/1102365176 Halverson [2004] Halverson T (2004) Representations of the q𝑞qitalic_q-rook monoid. J Algebra 273(1):227–251. 10.1016/j.jalgebra.2003.11.002, URL http://dx.doi.org/10.1016/j.jalgebra.2003.11.002 Halverson and delMas [2014] Halverson T, delMas E (2014) Representations of the Rook-Brauer algebra. Comm Algebra 42(1):423–443. 10.1080/00927872.2012.716120, URL https://doi.org/10.1080/00927872.2012.716120 Halverson and Ram [2005] Halverson T, Ram A (2005) Partition algebras. European J Combin 26(6):869–921. 10.1016/j.ejc.2004.06.005, URL http://dx.doi.org/10.1016/j.ejc.2004.06.005 Jacobson [1989] Jacobson N (1989) Basic algebra. II, 2nd edn. W. H. Freeman and Company, New York Jones [1994] Jones VFR (1994) The Potts model and the symmetric group. In: Subfactors (Kyuzeso, 1993). World Sci. Publ., River Edge, NJ, p 259–267 Kudryavtseva and Mazorchuk [2008] Kudryavtseva G, Mazorchuk V (2008) Schur-Weyl dualities for symmetric inverse semigroups. J Pure Appl Algebra 212(8):1987–1995. 10.1016/j.jpaa.2007.12.004, URL https://doi.org/10.1016/j.jpaa.2007.12.004 Lawson [1998] Lawson MV (1998) Inverse semigroups. World Scientific Publishing Co. Inc., River Edge, NJ, 10.1142/9789812816689, URL http://dx.doi.org/10.1142/9789812816689, the theory of partial symmetries Martin [1994] Martin P (1994) Temperley-Lieb algebras for nonplanar statistical mechanics—the partition algebra construction. J Knot Theory Ramifications 3(1):51–82. 10.1142/S0218216594000071, URL http://dx.doi.org/10.1142/S0218216594000071 Martin [1996] Martin P (1996) The structure of the partition algebras. J Algebra 183(2):319–358. 10.1006/jabr.1996.0223, URL http://dx.doi.org/10.1006/jabr.1996.0223 Martin and Mazorchuk [2014] Martin P, Mazorchuk V (2014) On the representation theory of partial Brauer algebras. Q J Math 65(1):225–247. 10.1093/qmath/has043, URL https://doi.org/10.1093/qmath/has043 Martin [1990] Martin PP (1990) Representations of graph Temperley-Lieb algebras. Publ Res Inst Math Sci 26(3):485–503. 10.2977/prims/1195170958, URL https://doi.org/10.2977/prims/1195170958 Martin [2008] Martin S (2008) Schur algebras and representation theory, Cambridge Tracts in Mathematics, vol 112. Cambridge University Press, Cambridge, reprint of the 1993 original Munn [1955] Munn WD (1955) On semigroup algebras. Proc Cambridge Philos Soc 51:1–15 Munn [1957a] Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Halverson T (1996) Characters of the centralizer algebras of mixed tensor representations of Gl(r,ℂ)Gl𝑟ℂ{\rm Gl}(r,\mathbb{C})roman_Gl ( italic_r , blackboard_C ) and the quantum group 𝔘q(ℊl(r,ℂ))subscript𝔘𝑞ℊ𝑙𝑟ℂ\mathfrak{U}_{q}({\mathcal{g}l}(r,\mathbb{C}))fraktur_U start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ( caligraphic_g italic_l ( italic_r , blackboard_C ) ). Pacific J Math 174(2):359–410. URL http://projecteuclid.org/euclid.pjm/1102365176 Halverson [2004] Halverson T (2004) Representations of the q𝑞qitalic_q-rook monoid. J Algebra 273(1):227–251. 10.1016/j.jalgebra.2003.11.002, URL http://dx.doi.org/10.1016/j.jalgebra.2003.11.002 Halverson and delMas [2014] Halverson T, delMas E (2014) Representations of the Rook-Brauer algebra. Comm Algebra 42(1):423–443. 10.1080/00927872.2012.716120, URL https://doi.org/10.1080/00927872.2012.716120 Halverson and Ram [2005] Halverson T, Ram A (2005) Partition algebras. European J Combin 26(6):869–921. 10.1016/j.ejc.2004.06.005, URL http://dx.doi.org/10.1016/j.ejc.2004.06.005 Jacobson [1989] Jacobson N (1989) Basic algebra. II, 2nd edn. W. H. Freeman and Company, New York Jones [1994] Jones VFR (1994) The Potts model and the symmetric group. In: Subfactors (Kyuzeso, 1993). World Sci. Publ., River Edge, NJ, p 259–267 Kudryavtseva and Mazorchuk [2008] Kudryavtseva G, Mazorchuk V (2008) Schur-Weyl dualities for symmetric inverse semigroups. J Pure Appl Algebra 212(8):1987–1995. 10.1016/j.jpaa.2007.12.004, URL https://doi.org/10.1016/j.jpaa.2007.12.004 Lawson [1998] Lawson MV (1998) Inverse semigroups. World Scientific Publishing Co. Inc., River Edge, NJ, 10.1142/9789812816689, URL http://dx.doi.org/10.1142/9789812816689, the theory of partial symmetries Martin [1994] Martin P (1994) Temperley-Lieb algebras for nonplanar statistical mechanics—the partition algebra construction. J Knot Theory Ramifications 3(1):51–82. 10.1142/S0218216594000071, URL http://dx.doi.org/10.1142/S0218216594000071 Martin [1996] Martin P (1996) The structure of the partition algebras. J Algebra 183(2):319–358. 10.1006/jabr.1996.0223, URL http://dx.doi.org/10.1006/jabr.1996.0223 Martin and Mazorchuk [2014] Martin P, Mazorchuk V (2014) On the representation theory of partial Brauer algebras. Q J Math 65(1):225–247. 10.1093/qmath/has043, URL https://doi.org/10.1093/qmath/has043 Martin [1990] Martin PP (1990) Representations of graph Temperley-Lieb algebras. Publ Res Inst Math Sci 26(3):485–503. 10.2977/prims/1195170958, URL https://doi.org/10.2977/prims/1195170958 Martin [2008] Martin S (2008) Schur algebras and representation theory, Cambridge Tracts in Mathematics, vol 112. Cambridge University Press, Cambridge, reprint of the 1993 original Munn [1955] Munn WD (1955) On semigroup algebras. Proc Cambridge Philos Soc 51:1–15 Munn [1957a] Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Halverson T (2004) Representations of the q𝑞qitalic_q-rook monoid. J Algebra 273(1):227–251. 10.1016/j.jalgebra.2003.11.002, URL http://dx.doi.org/10.1016/j.jalgebra.2003.11.002 Halverson and delMas [2014] Halverson T, delMas E (2014) Representations of the Rook-Brauer algebra. Comm Algebra 42(1):423–443. 10.1080/00927872.2012.716120, URL https://doi.org/10.1080/00927872.2012.716120 Halverson and Ram [2005] Halverson T, Ram A (2005) Partition algebras. European J Combin 26(6):869–921. 10.1016/j.ejc.2004.06.005, URL http://dx.doi.org/10.1016/j.ejc.2004.06.005 Jacobson [1989] Jacobson N (1989) Basic algebra. II, 2nd edn. W. H. Freeman and Company, New York Jones [1994] Jones VFR (1994) The Potts model and the symmetric group. In: Subfactors (Kyuzeso, 1993). World Sci. Publ., River Edge, NJ, p 259–267 Kudryavtseva and Mazorchuk [2008] Kudryavtseva G, Mazorchuk V (2008) Schur-Weyl dualities for symmetric inverse semigroups. J Pure Appl Algebra 212(8):1987–1995. 10.1016/j.jpaa.2007.12.004, URL https://doi.org/10.1016/j.jpaa.2007.12.004 Lawson [1998] Lawson MV (1998) Inverse semigroups. World Scientific Publishing Co. Inc., River Edge, NJ, 10.1142/9789812816689, URL http://dx.doi.org/10.1142/9789812816689, the theory of partial symmetries Martin [1994] Martin P (1994) Temperley-Lieb algebras for nonplanar statistical mechanics—the partition algebra construction. J Knot Theory Ramifications 3(1):51–82. 10.1142/S0218216594000071, URL http://dx.doi.org/10.1142/S0218216594000071 Martin [1996] Martin P (1996) The structure of the partition algebras. J Algebra 183(2):319–358. 10.1006/jabr.1996.0223, URL http://dx.doi.org/10.1006/jabr.1996.0223 Martin and Mazorchuk [2014] Martin P, Mazorchuk V (2014) On the representation theory of partial Brauer algebras. Q J Math 65(1):225–247. 10.1093/qmath/has043, URL https://doi.org/10.1093/qmath/has043 Martin [1990] Martin PP (1990) Representations of graph Temperley-Lieb algebras. Publ Res Inst Math Sci 26(3):485–503. 10.2977/prims/1195170958, URL https://doi.org/10.2977/prims/1195170958 Martin [2008] Martin S (2008) Schur algebras and representation theory, Cambridge Tracts in Mathematics, vol 112. Cambridge University Press, Cambridge, reprint of the 1993 original Munn [1955] Munn WD (1955) On semigroup algebras. Proc Cambridge Philos Soc 51:1–15 Munn [1957a] Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Halverson T, delMas E (2014) Representations of the Rook-Brauer algebra. Comm Algebra 42(1):423–443. 10.1080/00927872.2012.716120, URL https://doi.org/10.1080/00927872.2012.716120 Halverson and Ram [2005] Halverson T, Ram A (2005) Partition algebras. European J Combin 26(6):869–921. 10.1016/j.ejc.2004.06.005, URL http://dx.doi.org/10.1016/j.ejc.2004.06.005 Jacobson [1989] Jacobson N (1989) Basic algebra. II, 2nd edn. W. H. Freeman and Company, New York Jones [1994] Jones VFR (1994) The Potts model and the symmetric group. In: Subfactors (Kyuzeso, 1993). World Sci. Publ., River Edge, NJ, p 259–267 Kudryavtseva and Mazorchuk [2008] Kudryavtseva G, Mazorchuk V (2008) Schur-Weyl dualities for symmetric inverse semigroups. J Pure Appl Algebra 212(8):1987–1995. 10.1016/j.jpaa.2007.12.004, URL https://doi.org/10.1016/j.jpaa.2007.12.004 Lawson [1998] Lawson MV (1998) Inverse semigroups. World Scientific Publishing Co. Inc., River Edge, NJ, 10.1142/9789812816689, URL http://dx.doi.org/10.1142/9789812816689, the theory of partial symmetries Martin [1994] Martin P (1994) Temperley-Lieb algebras for nonplanar statistical mechanics—the partition algebra construction. J Knot Theory Ramifications 3(1):51–82. 10.1142/S0218216594000071, URL http://dx.doi.org/10.1142/S0218216594000071 Martin [1996] Martin P (1996) The structure of the partition algebras. J Algebra 183(2):319–358. 10.1006/jabr.1996.0223, URL http://dx.doi.org/10.1006/jabr.1996.0223 Martin and Mazorchuk [2014] Martin P, Mazorchuk V (2014) On the representation theory of partial Brauer algebras. Q J Math 65(1):225–247. 10.1093/qmath/has043, URL https://doi.org/10.1093/qmath/has043 Martin [1990] Martin PP (1990) Representations of graph Temperley-Lieb algebras. Publ Res Inst Math Sci 26(3):485–503. 10.2977/prims/1195170958, URL https://doi.org/10.2977/prims/1195170958 Martin [2008] Martin S (2008) Schur algebras and representation theory, Cambridge Tracts in Mathematics, vol 112. Cambridge University Press, Cambridge, reprint of the 1993 original Munn [1955] Munn WD (1955) On semigroup algebras. Proc Cambridge Philos Soc 51:1–15 Munn [1957a] Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Halverson T, Ram A (2005) Partition algebras. European J Combin 26(6):869–921. 10.1016/j.ejc.2004.06.005, URL http://dx.doi.org/10.1016/j.ejc.2004.06.005 Jacobson [1989] Jacobson N (1989) Basic algebra. II, 2nd edn. W. H. Freeman and Company, New York Jones [1994] Jones VFR (1994) The Potts model and the symmetric group. In: Subfactors (Kyuzeso, 1993). World Sci. Publ., River Edge, NJ, p 259–267 Kudryavtseva and Mazorchuk [2008] Kudryavtseva G, Mazorchuk V (2008) Schur-Weyl dualities for symmetric inverse semigroups. J Pure Appl Algebra 212(8):1987–1995. 10.1016/j.jpaa.2007.12.004, URL https://doi.org/10.1016/j.jpaa.2007.12.004 Lawson [1998] Lawson MV (1998) Inverse semigroups. World Scientific Publishing Co. Inc., River Edge, NJ, 10.1142/9789812816689, URL http://dx.doi.org/10.1142/9789812816689, the theory of partial symmetries Martin [1994] Martin P (1994) Temperley-Lieb algebras for nonplanar statistical mechanics—the partition algebra construction. J Knot Theory Ramifications 3(1):51–82. 10.1142/S0218216594000071, URL http://dx.doi.org/10.1142/S0218216594000071 Martin [1996] Martin P (1996) The structure of the partition algebras. J Algebra 183(2):319–358. 10.1006/jabr.1996.0223, URL http://dx.doi.org/10.1006/jabr.1996.0223 Martin and Mazorchuk [2014] Martin P, Mazorchuk V (2014) On the representation theory of partial Brauer algebras. Q J Math 65(1):225–247. 10.1093/qmath/has043, URL https://doi.org/10.1093/qmath/has043 Martin [1990] Martin PP (1990) Representations of graph Temperley-Lieb algebras. Publ Res Inst Math Sci 26(3):485–503. 10.2977/prims/1195170958, URL https://doi.org/10.2977/prims/1195170958 Martin [2008] Martin S (2008) Schur algebras and representation theory, Cambridge Tracts in Mathematics, vol 112. Cambridge University Press, Cambridge, reprint of the 1993 original Munn [1955] Munn WD (1955) On semigroup algebras. Proc Cambridge Philos Soc 51:1–15 Munn [1957a] Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Jacobson N (1989) Basic algebra. II, 2nd edn. W. H. Freeman and Company, New York Jones [1994] Jones VFR (1994) The Potts model and the symmetric group. In: Subfactors (Kyuzeso, 1993). World Sci. Publ., River Edge, NJ, p 259–267 Kudryavtseva and Mazorchuk [2008] Kudryavtseva G, Mazorchuk V (2008) Schur-Weyl dualities for symmetric inverse semigroups. J Pure Appl Algebra 212(8):1987–1995. 10.1016/j.jpaa.2007.12.004, URL https://doi.org/10.1016/j.jpaa.2007.12.004 Lawson [1998] Lawson MV (1998) Inverse semigroups. World Scientific Publishing Co. Inc., River Edge, NJ, 10.1142/9789812816689, URL http://dx.doi.org/10.1142/9789812816689, the theory of partial symmetries Martin [1994] Martin P (1994) Temperley-Lieb algebras for nonplanar statistical mechanics—the partition algebra construction. J Knot Theory Ramifications 3(1):51–82. 10.1142/S0218216594000071, URL http://dx.doi.org/10.1142/S0218216594000071 Martin [1996] Martin P (1996) The structure of the partition algebras. J Algebra 183(2):319–358. 10.1006/jabr.1996.0223, URL http://dx.doi.org/10.1006/jabr.1996.0223 Martin and Mazorchuk [2014] Martin P, Mazorchuk V (2014) On the representation theory of partial Brauer algebras. Q J Math 65(1):225–247. 10.1093/qmath/has043, URL https://doi.org/10.1093/qmath/has043 Martin [1990] Martin PP (1990) Representations of graph Temperley-Lieb algebras. Publ Res Inst Math Sci 26(3):485–503. 10.2977/prims/1195170958, URL https://doi.org/10.2977/prims/1195170958 Martin [2008] Martin S (2008) Schur algebras and representation theory, Cambridge Tracts in Mathematics, vol 112. Cambridge University Press, Cambridge, reprint of the 1993 original Munn [1955] Munn WD (1955) On semigroup algebras. Proc Cambridge Philos Soc 51:1–15 Munn [1957a] Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Jones VFR (1994) The Potts model and the symmetric group. In: Subfactors (Kyuzeso, 1993). World Sci. Publ., River Edge, NJ, p 259–267 Kudryavtseva and Mazorchuk [2008] Kudryavtseva G, Mazorchuk V (2008) Schur-Weyl dualities for symmetric inverse semigroups. J Pure Appl Algebra 212(8):1987–1995. 10.1016/j.jpaa.2007.12.004, URL https://doi.org/10.1016/j.jpaa.2007.12.004 Lawson [1998] Lawson MV (1998) Inverse semigroups. World Scientific Publishing Co. Inc., River Edge, NJ, 10.1142/9789812816689, URL http://dx.doi.org/10.1142/9789812816689, the theory of partial symmetries Martin [1994] Martin P (1994) Temperley-Lieb algebras for nonplanar statistical mechanics—the partition algebra construction. J Knot Theory Ramifications 3(1):51–82. 10.1142/S0218216594000071, URL http://dx.doi.org/10.1142/S0218216594000071 Martin [1996] Martin P (1996) The structure of the partition algebras. J Algebra 183(2):319–358. 10.1006/jabr.1996.0223, URL http://dx.doi.org/10.1006/jabr.1996.0223 Martin and Mazorchuk [2014] Martin P, Mazorchuk V (2014) On the representation theory of partial Brauer algebras. Q J Math 65(1):225–247. 10.1093/qmath/has043, URL https://doi.org/10.1093/qmath/has043 Martin [1990] Martin PP (1990) Representations of graph Temperley-Lieb algebras. Publ Res Inst Math Sci 26(3):485–503. 10.2977/prims/1195170958, URL https://doi.org/10.2977/prims/1195170958 Martin [2008] Martin S (2008) Schur algebras and representation theory, Cambridge Tracts in Mathematics, vol 112. Cambridge University Press, Cambridge, reprint of the 1993 original Munn [1955] Munn WD (1955) On semigroup algebras. Proc Cambridge Philos Soc 51:1–15 Munn [1957a] Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Kudryavtseva G, Mazorchuk V (2008) Schur-Weyl dualities for symmetric inverse semigroups. J Pure Appl Algebra 212(8):1987–1995. 10.1016/j.jpaa.2007.12.004, URL https://doi.org/10.1016/j.jpaa.2007.12.004 Lawson [1998] Lawson MV (1998) Inverse semigroups. World Scientific Publishing Co. Inc., River Edge, NJ, 10.1142/9789812816689, URL http://dx.doi.org/10.1142/9789812816689, the theory of partial symmetries Martin [1994] Martin P (1994) Temperley-Lieb algebras for nonplanar statistical mechanics—the partition algebra construction. J Knot Theory Ramifications 3(1):51–82. 10.1142/S0218216594000071, URL http://dx.doi.org/10.1142/S0218216594000071 Martin [1996] Martin P (1996) The structure of the partition algebras. J Algebra 183(2):319–358. 10.1006/jabr.1996.0223, URL http://dx.doi.org/10.1006/jabr.1996.0223 Martin and Mazorchuk [2014] Martin P, Mazorchuk V (2014) On the representation theory of partial Brauer algebras. Q J Math 65(1):225–247. 10.1093/qmath/has043, URL https://doi.org/10.1093/qmath/has043 Martin [1990] Martin PP (1990) Representations of graph Temperley-Lieb algebras. Publ Res Inst Math Sci 26(3):485–503. 10.2977/prims/1195170958, URL https://doi.org/10.2977/prims/1195170958 Martin [2008] Martin S (2008) Schur algebras and representation theory, Cambridge Tracts in Mathematics, vol 112. Cambridge University Press, Cambridge, reprint of the 1993 original Munn [1955] Munn WD (1955) On semigroup algebras. Proc Cambridge Philos Soc 51:1–15 Munn [1957a] Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Lawson MV (1998) Inverse semigroups. World Scientific Publishing Co. Inc., River Edge, NJ, 10.1142/9789812816689, URL http://dx.doi.org/10.1142/9789812816689, the theory of partial symmetries Martin [1994] Martin P (1994) Temperley-Lieb algebras for nonplanar statistical mechanics—the partition algebra construction. J Knot Theory Ramifications 3(1):51–82. 10.1142/S0218216594000071, URL http://dx.doi.org/10.1142/S0218216594000071 Martin [1996] Martin P (1996) The structure of the partition algebras. J Algebra 183(2):319–358. 10.1006/jabr.1996.0223, URL http://dx.doi.org/10.1006/jabr.1996.0223 Martin and Mazorchuk [2014] Martin P, Mazorchuk V (2014) On the representation theory of partial Brauer algebras. Q J Math 65(1):225–247. 10.1093/qmath/has043, URL https://doi.org/10.1093/qmath/has043 Martin [1990] Martin PP (1990) Representations of graph Temperley-Lieb algebras. Publ Res Inst Math Sci 26(3):485–503. 10.2977/prims/1195170958, URL https://doi.org/10.2977/prims/1195170958 Martin [2008] Martin S (2008) Schur algebras and representation theory, Cambridge Tracts in Mathematics, vol 112. Cambridge University Press, Cambridge, reprint of the 1993 original Munn [1955] Munn WD (1955) On semigroup algebras. Proc Cambridge Philos Soc 51:1–15 Munn [1957a] Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Martin P (1994) Temperley-Lieb algebras for nonplanar statistical mechanics—the partition algebra construction. J Knot Theory Ramifications 3(1):51–82. 10.1142/S0218216594000071, URL http://dx.doi.org/10.1142/S0218216594000071 Martin [1996] Martin P (1996) The structure of the partition algebras. J Algebra 183(2):319–358. 10.1006/jabr.1996.0223, URL http://dx.doi.org/10.1006/jabr.1996.0223 Martin and Mazorchuk [2014] Martin P, Mazorchuk V (2014) On the representation theory of partial Brauer algebras. Q J Math 65(1):225–247. 10.1093/qmath/has043, URL https://doi.org/10.1093/qmath/has043 Martin [1990] Martin PP (1990) Representations of graph Temperley-Lieb algebras. Publ Res Inst Math Sci 26(3):485–503. 10.2977/prims/1195170958, URL https://doi.org/10.2977/prims/1195170958 Martin [2008] Martin S (2008) Schur algebras and representation theory, Cambridge Tracts in Mathematics, vol 112. Cambridge University Press, Cambridge, reprint of the 1993 original Munn [1955] Munn WD (1955) On semigroup algebras. Proc Cambridge Philos Soc 51:1–15 Munn [1957a] Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Martin P (1996) The structure of the partition algebras. J Algebra 183(2):319–358. 10.1006/jabr.1996.0223, URL http://dx.doi.org/10.1006/jabr.1996.0223 Martin and Mazorchuk [2014] Martin P, Mazorchuk V (2014) On the representation theory of partial Brauer algebras. Q J Math 65(1):225–247. 10.1093/qmath/has043, URL https://doi.org/10.1093/qmath/has043 Martin [1990] Martin PP (1990) Representations of graph Temperley-Lieb algebras. Publ Res Inst Math Sci 26(3):485–503. 10.2977/prims/1195170958, URL https://doi.org/10.2977/prims/1195170958 Martin [2008] Martin S (2008) Schur algebras and representation theory, Cambridge Tracts in Mathematics, vol 112. Cambridge University Press, Cambridge, reprint of the 1993 original Munn [1955] Munn WD (1955) On semigroup algebras. Proc Cambridge Philos Soc 51:1–15 Munn [1957a] Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Martin P, Mazorchuk V (2014) On the representation theory of partial Brauer algebras. Q J Math 65(1):225–247. 10.1093/qmath/has043, URL https://doi.org/10.1093/qmath/has043 Martin [1990] Martin PP (1990) Representations of graph Temperley-Lieb algebras. Publ Res Inst Math Sci 26(3):485–503. 10.2977/prims/1195170958, URL https://doi.org/10.2977/prims/1195170958 Martin [2008] Martin S (2008) Schur algebras and representation theory, Cambridge Tracts in Mathematics, vol 112. Cambridge University Press, Cambridge, reprint of the 1993 original Munn [1955] Munn WD (1955) On semigroup algebras. Proc Cambridge Philos Soc 51:1–15 Munn [1957a] Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Martin PP (1990) Representations of graph Temperley-Lieb algebras. Publ Res Inst Math Sci 26(3):485–503. 10.2977/prims/1195170958, URL https://doi.org/10.2977/prims/1195170958 Martin [2008] Martin S (2008) Schur algebras and representation theory, Cambridge Tracts in Mathematics, vol 112. Cambridge University Press, Cambridge, reprint of the 1993 original Munn [1955] Munn WD (1955) On semigroup algebras. Proc Cambridge Philos Soc 51:1–15 Munn [1957a] Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Martin S (2008) Schur algebras and representation theory, Cambridge Tracts in Mathematics, vol 112. Cambridge University Press, Cambridge, reprint of the 1993 original Munn [1955] Munn WD (1955) On semigroup algebras. Proc Cambridge Philos Soc 51:1–15 Munn [1957a] Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Munn WD (1955) On semigroup algebras. Proc Cambridge Philos Soc 51:1–15 Munn [1957a] Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J.
- Doty S, Hu J (2009) Schur-Weyl duality for orthogonal groups. Proc Lond Math Soc (3) 98(3):679–713. 10.1112/plms/pdn044, URL https://doi.org/10.1112/plms/pdn044 Goodman and Wallach [2009] Goodman R, Wallach NR (2009) Symmetry, representations, and invariants, Graduate Texts in Mathematics, vol 255. Springer, Dordrecht, 10.1007/978-0-387-79852-3, URL https://doi.org/10.1007/978-0-387-79852-3 Graham and Lehrer [1996] Graham JJ, Lehrer GI (1996) Cellular algebras. Invent Math 123(1):1–34. 10.1007/BF01232365, URL http://dx.doi.org/10.1007/BF01232365 Green [1980] Green JA (1980) Polynomial representations of GLnsubscriptGL𝑛{\rm GL}_{n}roman_GL start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT, Lecture Notes in Mathematics, vol 830. Springer-Verlag, Berlin Grood [2006] Grood C (2006) The rook partition algebra. J Combin Theory Ser A 113(2):325–351. 10.1016/j.jcta.2005.03.006, URL https://doi.org/10.1016/j.jcta.2005.03.006 Halverson [1996] Halverson T (1996) Characters of the centralizer algebras of mixed tensor representations of Gl(r,ℂ)Gl𝑟ℂ{\rm Gl}(r,\mathbb{C})roman_Gl ( italic_r , blackboard_C ) and the quantum group 𝔘q(ℊl(r,ℂ))subscript𝔘𝑞ℊ𝑙𝑟ℂ\mathfrak{U}_{q}({\mathcal{g}l}(r,\mathbb{C}))fraktur_U start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ( caligraphic_g italic_l ( italic_r , blackboard_C ) ). Pacific J Math 174(2):359–410. URL http://projecteuclid.org/euclid.pjm/1102365176 Halverson [2004] Halverson T (2004) Representations of the q𝑞qitalic_q-rook monoid. J Algebra 273(1):227–251. 10.1016/j.jalgebra.2003.11.002, URL http://dx.doi.org/10.1016/j.jalgebra.2003.11.002 Halverson and delMas [2014] Halverson T, delMas E (2014) Representations of the Rook-Brauer algebra. Comm Algebra 42(1):423–443. 10.1080/00927872.2012.716120, URL https://doi.org/10.1080/00927872.2012.716120 Halverson and Ram [2005] Halverson T, Ram A (2005) Partition algebras. European J Combin 26(6):869–921. 10.1016/j.ejc.2004.06.005, URL http://dx.doi.org/10.1016/j.ejc.2004.06.005 Jacobson [1989] Jacobson N (1989) Basic algebra. II, 2nd edn. W. H. Freeman and Company, New York Jones [1994] Jones VFR (1994) The Potts model and the symmetric group. In: Subfactors (Kyuzeso, 1993). World Sci. Publ., River Edge, NJ, p 259–267 Kudryavtseva and Mazorchuk [2008] Kudryavtseva G, Mazorchuk V (2008) Schur-Weyl dualities for symmetric inverse semigroups. J Pure Appl Algebra 212(8):1987–1995. 10.1016/j.jpaa.2007.12.004, URL https://doi.org/10.1016/j.jpaa.2007.12.004 Lawson [1998] Lawson MV (1998) Inverse semigroups. World Scientific Publishing Co. Inc., River Edge, NJ, 10.1142/9789812816689, URL http://dx.doi.org/10.1142/9789812816689, the theory of partial symmetries Martin [1994] Martin P (1994) Temperley-Lieb algebras for nonplanar statistical mechanics—the partition algebra construction. J Knot Theory Ramifications 3(1):51–82. 10.1142/S0218216594000071, URL http://dx.doi.org/10.1142/S0218216594000071 Martin [1996] Martin P (1996) The structure of the partition algebras. J Algebra 183(2):319–358. 10.1006/jabr.1996.0223, URL http://dx.doi.org/10.1006/jabr.1996.0223 Martin and Mazorchuk [2014] Martin P, Mazorchuk V (2014) On the representation theory of partial Brauer algebras. Q J Math 65(1):225–247. 10.1093/qmath/has043, URL https://doi.org/10.1093/qmath/has043 Martin [1990] Martin PP (1990) Representations of graph Temperley-Lieb algebras. Publ Res Inst Math Sci 26(3):485–503. 10.2977/prims/1195170958, URL https://doi.org/10.2977/prims/1195170958 Martin [2008] Martin S (2008) Schur algebras and representation theory, Cambridge Tracts in Mathematics, vol 112. Cambridge University Press, Cambridge, reprint of the 1993 original Munn [1955] Munn WD (1955) On semigroup algebras. Proc Cambridge Philos Soc 51:1–15 Munn [1957a] Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Goodman R, Wallach NR (2009) Symmetry, representations, and invariants, Graduate Texts in Mathematics, vol 255. Springer, Dordrecht, 10.1007/978-0-387-79852-3, URL https://doi.org/10.1007/978-0-387-79852-3 Graham and Lehrer [1996] Graham JJ, Lehrer GI (1996) Cellular algebras. Invent Math 123(1):1–34. 10.1007/BF01232365, URL http://dx.doi.org/10.1007/BF01232365 Green [1980] Green JA (1980) Polynomial representations of GLnsubscriptGL𝑛{\rm GL}_{n}roman_GL start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT, Lecture Notes in Mathematics, vol 830. Springer-Verlag, Berlin Grood [2006] Grood C (2006) The rook partition algebra. J Combin Theory Ser A 113(2):325–351. 10.1016/j.jcta.2005.03.006, URL https://doi.org/10.1016/j.jcta.2005.03.006 Halverson [1996] Halverson T (1996) Characters of the centralizer algebras of mixed tensor representations of Gl(r,ℂ)Gl𝑟ℂ{\rm Gl}(r,\mathbb{C})roman_Gl ( italic_r , blackboard_C ) and the quantum group 𝔘q(ℊl(r,ℂ))subscript𝔘𝑞ℊ𝑙𝑟ℂ\mathfrak{U}_{q}({\mathcal{g}l}(r,\mathbb{C}))fraktur_U start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ( caligraphic_g italic_l ( italic_r , blackboard_C ) ). Pacific J Math 174(2):359–410. URL http://projecteuclid.org/euclid.pjm/1102365176 Halverson [2004] Halverson T (2004) Representations of the q𝑞qitalic_q-rook monoid. J Algebra 273(1):227–251. 10.1016/j.jalgebra.2003.11.002, URL http://dx.doi.org/10.1016/j.jalgebra.2003.11.002 Halverson and delMas [2014] Halverson T, delMas E (2014) Representations of the Rook-Brauer algebra. Comm Algebra 42(1):423–443. 10.1080/00927872.2012.716120, URL https://doi.org/10.1080/00927872.2012.716120 Halverson and Ram [2005] Halverson T, Ram A (2005) Partition algebras. European J Combin 26(6):869–921. 10.1016/j.ejc.2004.06.005, URL http://dx.doi.org/10.1016/j.ejc.2004.06.005 Jacobson [1989] Jacobson N (1989) Basic algebra. II, 2nd edn. W. H. Freeman and Company, New York Jones [1994] Jones VFR (1994) The Potts model and the symmetric group. In: Subfactors (Kyuzeso, 1993). World Sci. Publ., River Edge, NJ, p 259–267 Kudryavtseva and Mazorchuk [2008] Kudryavtseva G, Mazorchuk V (2008) Schur-Weyl dualities for symmetric inverse semigroups. J Pure Appl Algebra 212(8):1987–1995. 10.1016/j.jpaa.2007.12.004, URL https://doi.org/10.1016/j.jpaa.2007.12.004 Lawson [1998] Lawson MV (1998) Inverse semigroups. World Scientific Publishing Co. Inc., River Edge, NJ, 10.1142/9789812816689, URL http://dx.doi.org/10.1142/9789812816689, the theory of partial symmetries Martin [1994] Martin P (1994) Temperley-Lieb algebras for nonplanar statistical mechanics—the partition algebra construction. J Knot Theory Ramifications 3(1):51–82. 10.1142/S0218216594000071, URL http://dx.doi.org/10.1142/S0218216594000071 Martin [1996] Martin P (1996) The structure of the partition algebras. J Algebra 183(2):319–358. 10.1006/jabr.1996.0223, URL http://dx.doi.org/10.1006/jabr.1996.0223 Martin and Mazorchuk [2014] Martin P, Mazorchuk V (2014) On the representation theory of partial Brauer algebras. Q J Math 65(1):225–247. 10.1093/qmath/has043, URL https://doi.org/10.1093/qmath/has043 Martin [1990] Martin PP (1990) Representations of graph Temperley-Lieb algebras. Publ Res Inst Math Sci 26(3):485–503. 10.2977/prims/1195170958, URL https://doi.org/10.2977/prims/1195170958 Martin [2008] Martin S (2008) Schur algebras and representation theory, Cambridge Tracts in Mathematics, vol 112. Cambridge University Press, Cambridge, reprint of the 1993 original Munn [1955] Munn WD (1955) On semigroup algebras. Proc Cambridge Philos Soc 51:1–15 Munn [1957a] Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Graham JJ, Lehrer GI (1996) Cellular algebras. Invent Math 123(1):1–34. 10.1007/BF01232365, URL http://dx.doi.org/10.1007/BF01232365 Green [1980] Green JA (1980) Polynomial representations of GLnsubscriptGL𝑛{\rm GL}_{n}roman_GL start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT, Lecture Notes in Mathematics, vol 830. Springer-Verlag, Berlin Grood [2006] Grood C (2006) The rook partition algebra. J Combin Theory Ser A 113(2):325–351. 10.1016/j.jcta.2005.03.006, URL https://doi.org/10.1016/j.jcta.2005.03.006 Halverson [1996] Halverson T (1996) Characters of the centralizer algebras of mixed tensor representations of Gl(r,ℂ)Gl𝑟ℂ{\rm Gl}(r,\mathbb{C})roman_Gl ( italic_r , blackboard_C ) and the quantum group 𝔘q(ℊl(r,ℂ))subscript𝔘𝑞ℊ𝑙𝑟ℂ\mathfrak{U}_{q}({\mathcal{g}l}(r,\mathbb{C}))fraktur_U start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ( caligraphic_g italic_l ( italic_r , blackboard_C ) ). Pacific J Math 174(2):359–410. URL http://projecteuclid.org/euclid.pjm/1102365176 Halverson [2004] Halverson T (2004) Representations of the q𝑞qitalic_q-rook monoid. J Algebra 273(1):227–251. 10.1016/j.jalgebra.2003.11.002, URL http://dx.doi.org/10.1016/j.jalgebra.2003.11.002 Halverson and delMas [2014] Halverson T, delMas E (2014) Representations of the Rook-Brauer algebra. Comm Algebra 42(1):423–443. 10.1080/00927872.2012.716120, URL https://doi.org/10.1080/00927872.2012.716120 Halverson and Ram [2005] Halverson T, Ram A (2005) Partition algebras. European J Combin 26(6):869–921. 10.1016/j.ejc.2004.06.005, URL http://dx.doi.org/10.1016/j.ejc.2004.06.005 Jacobson [1989] Jacobson N (1989) Basic algebra. II, 2nd edn. W. H. Freeman and Company, New York Jones [1994] Jones VFR (1994) The Potts model and the symmetric group. In: Subfactors (Kyuzeso, 1993). World Sci. Publ., River Edge, NJ, p 259–267 Kudryavtseva and Mazorchuk [2008] Kudryavtseva G, Mazorchuk V (2008) Schur-Weyl dualities for symmetric inverse semigroups. J Pure Appl Algebra 212(8):1987–1995. 10.1016/j.jpaa.2007.12.004, URL https://doi.org/10.1016/j.jpaa.2007.12.004 Lawson [1998] Lawson MV (1998) Inverse semigroups. World Scientific Publishing Co. Inc., River Edge, NJ, 10.1142/9789812816689, URL http://dx.doi.org/10.1142/9789812816689, the theory of partial symmetries Martin [1994] Martin P (1994) Temperley-Lieb algebras for nonplanar statistical mechanics—the partition algebra construction. J Knot Theory Ramifications 3(1):51–82. 10.1142/S0218216594000071, URL http://dx.doi.org/10.1142/S0218216594000071 Martin [1996] Martin P (1996) The structure of the partition algebras. J Algebra 183(2):319–358. 10.1006/jabr.1996.0223, URL http://dx.doi.org/10.1006/jabr.1996.0223 Martin and Mazorchuk [2014] Martin P, Mazorchuk V (2014) On the representation theory of partial Brauer algebras. Q J Math 65(1):225–247. 10.1093/qmath/has043, URL https://doi.org/10.1093/qmath/has043 Martin [1990] Martin PP (1990) Representations of graph Temperley-Lieb algebras. Publ Res Inst Math Sci 26(3):485–503. 10.2977/prims/1195170958, URL https://doi.org/10.2977/prims/1195170958 Martin [2008] Martin S (2008) Schur algebras and representation theory, Cambridge Tracts in Mathematics, vol 112. Cambridge University Press, Cambridge, reprint of the 1993 original Munn [1955] Munn WD (1955) On semigroup algebras. Proc Cambridge Philos Soc 51:1–15 Munn [1957a] Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Green JA (1980) Polynomial representations of GLnsubscriptGL𝑛{\rm GL}_{n}roman_GL start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT, Lecture Notes in Mathematics, vol 830. Springer-Verlag, Berlin Grood [2006] Grood C (2006) The rook partition algebra. J Combin Theory Ser A 113(2):325–351. 10.1016/j.jcta.2005.03.006, URL https://doi.org/10.1016/j.jcta.2005.03.006 Halverson [1996] Halverson T (1996) Characters of the centralizer algebras of mixed tensor representations of Gl(r,ℂ)Gl𝑟ℂ{\rm Gl}(r,\mathbb{C})roman_Gl ( italic_r , blackboard_C ) and the quantum group 𝔘q(ℊl(r,ℂ))subscript𝔘𝑞ℊ𝑙𝑟ℂ\mathfrak{U}_{q}({\mathcal{g}l}(r,\mathbb{C}))fraktur_U start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ( caligraphic_g italic_l ( italic_r , blackboard_C ) ). Pacific J Math 174(2):359–410. URL http://projecteuclid.org/euclid.pjm/1102365176 Halverson [2004] Halverson T (2004) Representations of the q𝑞qitalic_q-rook monoid. J Algebra 273(1):227–251. 10.1016/j.jalgebra.2003.11.002, URL http://dx.doi.org/10.1016/j.jalgebra.2003.11.002 Halverson and delMas [2014] Halverson T, delMas E (2014) Representations of the Rook-Brauer algebra. Comm Algebra 42(1):423–443. 10.1080/00927872.2012.716120, URL https://doi.org/10.1080/00927872.2012.716120 Halverson and Ram [2005] Halverson T, Ram A (2005) Partition algebras. European J Combin 26(6):869–921. 10.1016/j.ejc.2004.06.005, URL http://dx.doi.org/10.1016/j.ejc.2004.06.005 Jacobson [1989] Jacobson N (1989) Basic algebra. II, 2nd edn. W. H. Freeman and Company, New York Jones [1994] Jones VFR (1994) The Potts model and the symmetric group. In: Subfactors (Kyuzeso, 1993). World Sci. Publ., River Edge, NJ, p 259–267 Kudryavtseva and Mazorchuk [2008] Kudryavtseva G, Mazorchuk V (2008) Schur-Weyl dualities for symmetric inverse semigroups. J Pure Appl Algebra 212(8):1987–1995. 10.1016/j.jpaa.2007.12.004, URL https://doi.org/10.1016/j.jpaa.2007.12.004 Lawson [1998] Lawson MV (1998) Inverse semigroups. World Scientific Publishing Co. Inc., River Edge, NJ, 10.1142/9789812816689, URL http://dx.doi.org/10.1142/9789812816689, the theory of partial symmetries Martin [1994] Martin P (1994) Temperley-Lieb algebras for nonplanar statistical mechanics—the partition algebra construction. J Knot Theory Ramifications 3(1):51–82. 10.1142/S0218216594000071, URL http://dx.doi.org/10.1142/S0218216594000071 Martin [1996] Martin P (1996) The structure of the partition algebras. J Algebra 183(2):319–358. 10.1006/jabr.1996.0223, URL http://dx.doi.org/10.1006/jabr.1996.0223 Martin and Mazorchuk [2014] Martin P, Mazorchuk V (2014) On the representation theory of partial Brauer algebras. Q J Math 65(1):225–247. 10.1093/qmath/has043, URL https://doi.org/10.1093/qmath/has043 Martin [1990] Martin PP (1990) Representations of graph Temperley-Lieb algebras. Publ Res Inst Math Sci 26(3):485–503. 10.2977/prims/1195170958, URL https://doi.org/10.2977/prims/1195170958 Martin [2008] Martin S (2008) Schur algebras and representation theory, Cambridge Tracts in Mathematics, vol 112. Cambridge University Press, Cambridge, reprint of the 1993 original Munn [1955] Munn WD (1955) On semigroup algebras. Proc Cambridge Philos Soc 51:1–15 Munn [1957a] Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Grood C (2006) The rook partition algebra. J Combin Theory Ser A 113(2):325–351. 10.1016/j.jcta.2005.03.006, URL https://doi.org/10.1016/j.jcta.2005.03.006 Halverson [1996] Halverson T (1996) Characters of the centralizer algebras of mixed tensor representations of Gl(r,ℂ)Gl𝑟ℂ{\rm Gl}(r,\mathbb{C})roman_Gl ( italic_r , blackboard_C ) and the quantum group 𝔘q(ℊl(r,ℂ))subscript𝔘𝑞ℊ𝑙𝑟ℂ\mathfrak{U}_{q}({\mathcal{g}l}(r,\mathbb{C}))fraktur_U start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ( caligraphic_g italic_l ( italic_r , blackboard_C ) ). Pacific J Math 174(2):359–410. URL http://projecteuclid.org/euclid.pjm/1102365176 Halverson [2004] Halverson T (2004) Representations of the q𝑞qitalic_q-rook monoid. J Algebra 273(1):227–251. 10.1016/j.jalgebra.2003.11.002, URL http://dx.doi.org/10.1016/j.jalgebra.2003.11.002 Halverson and delMas [2014] Halverson T, delMas E (2014) Representations of the Rook-Brauer algebra. Comm Algebra 42(1):423–443. 10.1080/00927872.2012.716120, URL https://doi.org/10.1080/00927872.2012.716120 Halverson and Ram [2005] Halverson T, Ram A (2005) Partition algebras. European J Combin 26(6):869–921. 10.1016/j.ejc.2004.06.005, URL http://dx.doi.org/10.1016/j.ejc.2004.06.005 Jacobson [1989] Jacobson N (1989) Basic algebra. II, 2nd edn. W. H. Freeman and Company, New York Jones [1994] Jones VFR (1994) The Potts model and the symmetric group. In: Subfactors (Kyuzeso, 1993). World Sci. Publ., River Edge, NJ, p 259–267 Kudryavtseva and Mazorchuk [2008] Kudryavtseva G, Mazorchuk V (2008) Schur-Weyl dualities for symmetric inverse semigroups. J Pure Appl Algebra 212(8):1987–1995. 10.1016/j.jpaa.2007.12.004, URL https://doi.org/10.1016/j.jpaa.2007.12.004 Lawson [1998] Lawson MV (1998) Inverse semigroups. World Scientific Publishing Co. Inc., River Edge, NJ, 10.1142/9789812816689, URL http://dx.doi.org/10.1142/9789812816689, the theory of partial symmetries Martin [1994] Martin P (1994) Temperley-Lieb algebras for nonplanar statistical mechanics—the partition algebra construction. J Knot Theory Ramifications 3(1):51–82. 10.1142/S0218216594000071, URL http://dx.doi.org/10.1142/S0218216594000071 Martin [1996] Martin P (1996) The structure of the partition algebras. J Algebra 183(2):319–358. 10.1006/jabr.1996.0223, URL http://dx.doi.org/10.1006/jabr.1996.0223 Martin and Mazorchuk [2014] Martin P, Mazorchuk V (2014) On the representation theory of partial Brauer algebras. Q J Math 65(1):225–247. 10.1093/qmath/has043, URL https://doi.org/10.1093/qmath/has043 Martin [1990] Martin PP (1990) Representations of graph Temperley-Lieb algebras. Publ Res Inst Math Sci 26(3):485–503. 10.2977/prims/1195170958, URL https://doi.org/10.2977/prims/1195170958 Martin [2008] Martin S (2008) Schur algebras and representation theory, Cambridge Tracts in Mathematics, vol 112. Cambridge University Press, Cambridge, reprint of the 1993 original Munn [1955] Munn WD (1955) On semigroup algebras. Proc Cambridge Philos Soc 51:1–15 Munn [1957a] Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Halverson T (1996) Characters of the centralizer algebras of mixed tensor representations of Gl(r,ℂ)Gl𝑟ℂ{\rm Gl}(r,\mathbb{C})roman_Gl ( italic_r , blackboard_C ) and the quantum group 𝔘q(ℊl(r,ℂ))subscript𝔘𝑞ℊ𝑙𝑟ℂ\mathfrak{U}_{q}({\mathcal{g}l}(r,\mathbb{C}))fraktur_U start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ( caligraphic_g italic_l ( italic_r , blackboard_C ) ). Pacific J Math 174(2):359–410. URL http://projecteuclid.org/euclid.pjm/1102365176 Halverson [2004] Halverson T (2004) Representations of the q𝑞qitalic_q-rook monoid. J Algebra 273(1):227–251. 10.1016/j.jalgebra.2003.11.002, URL http://dx.doi.org/10.1016/j.jalgebra.2003.11.002 Halverson and delMas [2014] Halverson T, delMas E (2014) Representations of the Rook-Brauer algebra. Comm Algebra 42(1):423–443. 10.1080/00927872.2012.716120, URL https://doi.org/10.1080/00927872.2012.716120 Halverson and Ram [2005] Halverson T, Ram A (2005) Partition algebras. European J Combin 26(6):869–921. 10.1016/j.ejc.2004.06.005, URL http://dx.doi.org/10.1016/j.ejc.2004.06.005 Jacobson [1989] Jacobson N (1989) Basic algebra. II, 2nd edn. W. H. Freeman and Company, New York Jones [1994] Jones VFR (1994) The Potts model and the symmetric group. In: Subfactors (Kyuzeso, 1993). World Sci. Publ., River Edge, NJ, p 259–267 Kudryavtseva and Mazorchuk [2008] Kudryavtseva G, Mazorchuk V (2008) Schur-Weyl dualities for symmetric inverse semigroups. J Pure Appl Algebra 212(8):1987–1995. 10.1016/j.jpaa.2007.12.004, URL https://doi.org/10.1016/j.jpaa.2007.12.004 Lawson [1998] Lawson MV (1998) Inverse semigroups. World Scientific Publishing Co. Inc., River Edge, NJ, 10.1142/9789812816689, URL http://dx.doi.org/10.1142/9789812816689, the theory of partial symmetries Martin [1994] Martin P (1994) Temperley-Lieb algebras for nonplanar statistical mechanics—the partition algebra construction. J Knot Theory Ramifications 3(1):51–82. 10.1142/S0218216594000071, URL http://dx.doi.org/10.1142/S0218216594000071 Martin [1996] Martin P (1996) The structure of the partition algebras. J Algebra 183(2):319–358. 10.1006/jabr.1996.0223, URL http://dx.doi.org/10.1006/jabr.1996.0223 Martin and Mazorchuk [2014] Martin P, Mazorchuk V (2014) On the representation theory of partial Brauer algebras. Q J Math 65(1):225–247. 10.1093/qmath/has043, URL https://doi.org/10.1093/qmath/has043 Martin [1990] Martin PP (1990) Representations of graph Temperley-Lieb algebras. Publ Res Inst Math Sci 26(3):485–503. 10.2977/prims/1195170958, URL https://doi.org/10.2977/prims/1195170958 Martin [2008] Martin S (2008) Schur algebras and representation theory, Cambridge Tracts in Mathematics, vol 112. Cambridge University Press, Cambridge, reprint of the 1993 original Munn [1955] Munn WD (1955) On semigroup algebras. Proc Cambridge Philos Soc 51:1–15 Munn [1957a] Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Halverson T (2004) Representations of the q𝑞qitalic_q-rook monoid. J Algebra 273(1):227–251. 10.1016/j.jalgebra.2003.11.002, URL http://dx.doi.org/10.1016/j.jalgebra.2003.11.002 Halverson and delMas [2014] Halverson T, delMas E (2014) Representations of the Rook-Brauer algebra. Comm Algebra 42(1):423–443. 10.1080/00927872.2012.716120, URL https://doi.org/10.1080/00927872.2012.716120 Halverson and Ram [2005] Halverson T, Ram A (2005) Partition algebras. European J Combin 26(6):869–921. 10.1016/j.ejc.2004.06.005, URL http://dx.doi.org/10.1016/j.ejc.2004.06.005 Jacobson [1989] Jacobson N (1989) Basic algebra. II, 2nd edn. W. H. Freeman and Company, New York Jones [1994] Jones VFR (1994) The Potts model and the symmetric group. In: Subfactors (Kyuzeso, 1993). World Sci. Publ., River Edge, NJ, p 259–267 Kudryavtseva and Mazorchuk [2008] Kudryavtseva G, Mazorchuk V (2008) Schur-Weyl dualities for symmetric inverse semigroups. J Pure Appl Algebra 212(8):1987–1995. 10.1016/j.jpaa.2007.12.004, URL https://doi.org/10.1016/j.jpaa.2007.12.004 Lawson [1998] Lawson MV (1998) Inverse semigroups. World Scientific Publishing Co. Inc., River Edge, NJ, 10.1142/9789812816689, URL http://dx.doi.org/10.1142/9789812816689, the theory of partial symmetries Martin [1994] Martin P (1994) Temperley-Lieb algebras for nonplanar statistical mechanics—the partition algebra construction. J Knot Theory Ramifications 3(1):51–82. 10.1142/S0218216594000071, URL http://dx.doi.org/10.1142/S0218216594000071 Martin [1996] Martin P (1996) The structure of the partition algebras. J Algebra 183(2):319–358. 10.1006/jabr.1996.0223, URL http://dx.doi.org/10.1006/jabr.1996.0223 Martin and Mazorchuk [2014] Martin P, Mazorchuk V (2014) On the representation theory of partial Brauer algebras. Q J Math 65(1):225–247. 10.1093/qmath/has043, URL https://doi.org/10.1093/qmath/has043 Martin [1990] Martin PP (1990) Representations of graph Temperley-Lieb algebras. Publ Res Inst Math Sci 26(3):485–503. 10.2977/prims/1195170958, URL https://doi.org/10.2977/prims/1195170958 Martin [2008] Martin S (2008) Schur algebras and representation theory, Cambridge Tracts in Mathematics, vol 112. Cambridge University Press, Cambridge, reprint of the 1993 original Munn [1955] Munn WD (1955) On semigroup algebras. Proc Cambridge Philos Soc 51:1–15 Munn [1957a] Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Halverson T, delMas E (2014) Representations of the Rook-Brauer algebra. Comm Algebra 42(1):423–443. 10.1080/00927872.2012.716120, URL https://doi.org/10.1080/00927872.2012.716120 Halverson and Ram [2005] Halverson T, Ram A (2005) Partition algebras. European J Combin 26(6):869–921. 10.1016/j.ejc.2004.06.005, URL http://dx.doi.org/10.1016/j.ejc.2004.06.005 Jacobson [1989] Jacobson N (1989) Basic algebra. II, 2nd edn. W. H. Freeman and Company, New York Jones [1994] Jones VFR (1994) The Potts model and the symmetric group. In: Subfactors (Kyuzeso, 1993). World Sci. Publ., River Edge, NJ, p 259–267 Kudryavtseva and Mazorchuk [2008] Kudryavtseva G, Mazorchuk V (2008) Schur-Weyl dualities for symmetric inverse semigroups. J Pure Appl Algebra 212(8):1987–1995. 10.1016/j.jpaa.2007.12.004, URL https://doi.org/10.1016/j.jpaa.2007.12.004 Lawson [1998] Lawson MV (1998) Inverse semigroups. World Scientific Publishing Co. Inc., River Edge, NJ, 10.1142/9789812816689, URL http://dx.doi.org/10.1142/9789812816689, the theory of partial symmetries Martin [1994] Martin P (1994) Temperley-Lieb algebras for nonplanar statistical mechanics—the partition algebra construction. J Knot Theory Ramifications 3(1):51–82. 10.1142/S0218216594000071, URL http://dx.doi.org/10.1142/S0218216594000071 Martin [1996] Martin P (1996) The structure of the partition algebras. J Algebra 183(2):319–358. 10.1006/jabr.1996.0223, URL http://dx.doi.org/10.1006/jabr.1996.0223 Martin and Mazorchuk [2014] Martin P, Mazorchuk V (2014) On the representation theory of partial Brauer algebras. Q J Math 65(1):225–247. 10.1093/qmath/has043, URL https://doi.org/10.1093/qmath/has043 Martin [1990] Martin PP (1990) Representations of graph Temperley-Lieb algebras. Publ Res Inst Math Sci 26(3):485–503. 10.2977/prims/1195170958, URL https://doi.org/10.2977/prims/1195170958 Martin [2008] Martin S (2008) Schur algebras and representation theory, Cambridge Tracts in Mathematics, vol 112. Cambridge University Press, Cambridge, reprint of the 1993 original Munn [1955] Munn WD (1955) On semigroup algebras. Proc Cambridge Philos Soc 51:1–15 Munn [1957a] Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Halverson T, Ram A (2005) Partition algebras. European J Combin 26(6):869–921. 10.1016/j.ejc.2004.06.005, URL http://dx.doi.org/10.1016/j.ejc.2004.06.005 Jacobson [1989] Jacobson N (1989) Basic algebra. II, 2nd edn. W. H. Freeman and Company, New York Jones [1994] Jones VFR (1994) The Potts model and the symmetric group. In: Subfactors (Kyuzeso, 1993). World Sci. Publ., River Edge, NJ, p 259–267 Kudryavtseva and Mazorchuk [2008] Kudryavtseva G, Mazorchuk V (2008) Schur-Weyl dualities for symmetric inverse semigroups. J Pure Appl Algebra 212(8):1987–1995. 10.1016/j.jpaa.2007.12.004, URL https://doi.org/10.1016/j.jpaa.2007.12.004 Lawson [1998] Lawson MV (1998) Inverse semigroups. World Scientific Publishing Co. Inc., River Edge, NJ, 10.1142/9789812816689, URL http://dx.doi.org/10.1142/9789812816689, the theory of partial symmetries Martin [1994] Martin P (1994) Temperley-Lieb algebras for nonplanar statistical mechanics—the partition algebra construction. J Knot Theory Ramifications 3(1):51–82. 10.1142/S0218216594000071, URL http://dx.doi.org/10.1142/S0218216594000071 Martin [1996] Martin P (1996) The structure of the partition algebras. J Algebra 183(2):319–358. 10.1006/jabr.1996.0223, URL http://dx.doi.org/10.1006/jabr.1996.0223 Martin and Mazorchuk [2014] Martin P, Mazorchuk V (2014) On the representation theory of partial Brauer algebras. Q J Math 65(1):225–247. 10.1093/qmath/has043, URL https://doi.org/10.1093/qmath/has043 Martin [1990] Martin PP (1990) Representations of graph Temperley-Lieb algebras. Publ Res Inst Math Sci 26(3):485–503. 10.2977/prims/1195170958, URL https://doi.org/10.2977/prims/1195170958 Martin [2008] Martin S (2008) Schur algebras and representation theory, Cambridge Tracts in Mathematics, vol 112. Cambridge University Press, Cambridge, reprint of the 1993 original Munn [1955] Munn WD (1955) On semigroup algebras. Proc Cambridge Philos Soc 51:1–15 Munn [1957a] Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Jacobson N (1989) Basic algebra. II, 2nd edn. W. H. Freeman and Company, New York Jones [1994] Jones VFR (1994) The Potts model and the symmetric group. In: Subfactors (Kyuzeso, 1993). World Sci. Publ., River Edge, NJ, p 259–267 Kudryavtseva and Mazorchuk [2008] Kudryavtseva G, Mazorchuk V (2008) Schur-Weyl dualities for symmetric inverse semigroups. J Pure Appl Algebra 212(8):1987–1995. 10.1016/j.jpaa.2007.12.004, URL https://doi.org/10.1016/j.jpaa.2007.12.004 Lawson [1998] Lawson MV (1998) Inverse semigroups. World Scientific Publishing Co. Inc., River Edge, NJ, 10.1142/9789812816689, URL http://dx.doi.org/10.1142/9789812816689, the theory of partial symmetries Martin [1994] Martin P (1994) Temperley-Lieb algebras for nonplanar statistical mechanics—the partition algebra construction. J Knot Theory Ramifications 3(1):51–82. 10.1142/S0218216594000071, URL http://dx.doi.org/10.1142/S0218216594000071 Martin [1996] Martin P (1996) The structure of the partition algebras. J Algebra 183(2):319–358. 10.1006/jabr.1996.0223, URL http://dx.doi.org/10.1006/jabr.1996.0223 Martin and Mazorchuk [2014] Martin P, Mazorchuk V (2014) On the representation theory of partial Brauer algebras. Q J Math 65(1):225–247. 10.1093/qmath/has043, URL https://doi.org/10.1093/qmath/has043 Martin [1990] Martin PP (1990) Representations of graph Temperley-Lieb algebras. Publ Res Inst Math Sci 26(3):485–503. 10.2977/prims/1195170958, URL https://doi.org/10.2977/prims/1195170958 Martin [2008] Martin S (2008) Schur algebras and representation theory, Cambridge Tracts in Mathematics, vol 112. Cambridge University Press, Cambridge, reprint of the 1993 original Munn [1955] Munn WD (1955) On semigroup algebras. Proc Cambridge Philos Soc 51:1–15 Munn [1957a] Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Jones VFR (1994) The Potts model and the symmetric group. In: Subfactors (Kyuzeso, 1993). World Sci. Publ., River Edge, NJ, p 259–267 Kudryavtseva and Mazorchuk [2008] Kudryavtseva G, Mazorchuk V (2008) Schur-Weyl dualities for symmetric inverse semigroups. J Pure Appl Algebra 212(8):1987–1995. 10.1016/j.jpaa.2007.12.004, URL https://doi.org/10.1016/j.jpaa.2007.12.004 Lawson [1998] Lawson MV (1998) Inverse semigroups. World Scientific Publishing Co. Inc., River Edge, NJ, 10.1142/9789812816689, URL http://dx.doi.org/10.1142/9789812816689, the theory of partial symmetries Martin [1994] Martin P (1994) Temperley-Lieb algebras for nonplanar statistical mechanics—the partition algebra construction. J Knot Theory Ramifications 3(1):51–82. 10.1142/S0218216594000071, URL http://dx.doi.org/10.1142/S0218216594000071 Martin [1996] Martin P (1996) The structure of the partition algebras. J Algebra 183(2):319–358. 10.1006/jabr.1996.0223, URL http://dx.doi.org/10.1006/jabr.1996.0223 Martin and Mazorchuk [2014] Martin P, Mazorchuk V (2014) On the representation theory of partial Brauer algebras. Q J Math 65(1):225–247. 10.1093/qmath/has043, URL https://doi.org/10.1093/qmath/has043 Martin [1990] Martin PP (1990) Representations of graph Temperley-Lieb algebras. Publ Res Inst Math Sci 26(3):485–503. 10.2977/prims/1195170958, URL https://doi.org/10.2977/prims/1195170958 Martin [2008] Martin S (2008) Schur algebras and representation theory, Cambridge Tracts in Mathematics, vol 112. Cambridge University Press, Cambridge, reprint of the 1993 original Munn [1955] Munn WD (1955) On semigroup algebras. Proc Cambridge Philos Soc 51:1–15 Munn [1957a] Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Kudryavtseva G, Mazorchuk V (2008) Schur-Weyl dualities for symmetric inverse semigroups. J Pure Appl Algebra 212(8):1987–1995. 10.1016/j.jpaa.2007.12.004, URL https://doi.org/10.1016/j.jpaa.2007.12.004 Lawson [1998] Lawson MV (1998) Inverse semigroups. World Scientific Publishing Co. Inc., River Edge, NJ, 10.1142/9789812816689, URL http://dx.doi.org/10.1142/9789812816689, the theory of partial symmetries Martin [1994] Martin P (1994) Temperley-Lieb algebras for nonplanar statistical mechanics—the partition algebra construction. J Knot Theory Ramifications 3(1):51–82. 10.1142/S0218216594000071, URL http://dx.doi.org/10.1142/S0218216594000071 Martin [1996] Martin P (1996) The structure of the partition algebras. J Algebra 183(2):319–358. 10.1006/jabr.1996.0223, URL http://dx.doi.org/10.1006/jabr.1996.0223 Martin and Mazorchuk [2014] Martin P, Mazorchuk V (2014) On the representation theory of partial Brauer algebras. Q J Math 65(1):225–247. 10.1093/qmath/has043, URL https://doi.org/10.1093/qmath/has043 Martin [1990] Martin PP (1990) Representations of graph Temperley-Lieb algebras. Publ Res Inst Math Sci 26(3):485–503. 10.2977/prims/1195170958, URL https://doi.org/10.2977/prims/1195170958 Martin [2008] Martin S (2008) Schur algebras and representation theory, Cambridge Tracts in Mathematics, vol 112. Cambridge University Press, Cambridge, reprint of the 1993 original Munn [1955] Munn WD (1955) On semigroup algebras. Proc Cambridge Philos Soc 51:1–15 Munn [1957a] Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Lawson MV (1998) Inverse semigroups. World Scientific Publishing Co. Inc., River Edge, NJ, 10.1142/9789812816689, URL http://dx.doi.org/10.1142/9789812816689, the theory of partial symmetries Martin [1994] Martin P (1994) Temperley-Lieb algebras for nonplanar statistical mechanics—the partition algebra construction. J Knot Theory Ramifications 3(1):51–82. 10.1142/S0218216594000071, URL http://dx.doi.org/10.1142/S0218216594000071 Martin [1996] Martin P (1996) The structure of the partition algebras. J Algebra 183(2):319–358. 10.1006/jabr.1996.0223, URL http://dx.doi.org/10.1006/jabr.1996.0223 Martin and Mazorchuk [2014] Martin P, Mazorchuk V (2014) On the representation theory of partial Brauer algebras. Q J Math 65(1):225–247. 10.1093/qmath/has043, URL https://doi.org/10.1093/qmath/has043 Martin [1990] Martin PP (1990) Representations of graph Temperley-Lieb algebras. Publ Res Inst Math Sci 26(3):485–503. 10.2977/prims/1195170958, URL https://doi.org/10.2977/prims/1195170958 Martin [2008] Martin S (2008) Schur algebras and representation theory, Cambridge Tracts in Mathematics, vol 112. Cambridge University Press, Cambridge, reprint of the 1993 original Munn [1955] Munn WD (1955) On semigroup algebras. Proc Cambridge Philos Soc 51:1–15 Munn [1957a] Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Martin P (1994) Temperley-Lieb algebras for nonplanar statistical mechanics—the partition algebra construction. J Knot Theory Ramifications 3(1):51–82. 10.1142/S0218216594000071, URL http://dx.doi.org/10.1142/S0218216594000071 Martin [1996] Martin P (1996) The structure of the partition algebras. J Algebra 183(2):319–358. 10.1006/jabr.1996.0223, URL http://dx.doi.org/10.1006/jabr.1996.0223 Martin and Mazorchuk [2014] Martin P, Mazorchuk V (2014) On the representation theory of partial Brauer algebras. Q J Math 65(1):225–247. 10.1093/qmath/has043, URL https://doi.org/10.1093/qmath/has043 Martin [1990] Martin PP (1990) Representations of graph Temperley-Lieb algebras. Publ Res Inst Math Sci 26(3):485–503. 10.2977/prims/1195170958, URL https://doi.org/10.2977/prims/1195170958 Martin [2008] Martin S (2008) Schur algebras and representation theory, Cambridge Tracts in Mathematics, vol 112. Cambridge University Press, Cambridge, reprint of the 1993 original Munn [1955] Munn WD (1955) On semigroup algebras. Proc Cambridge Philos Soc 51:1–15 Munn [1957a] Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Martin P (1996) The structure of the partition algebras. J Algebra 183(2):319–358. 10.1006/jabr.1996.0223, URL http://dx.doi.org/10.1006/jabr.1996.0223 Martin and Mazorchuk [2014] Martin P, Mazorchuk V (2014) On the representation theory of partial Brauer algebras. Q J Math 65(1):225–247. 10.1093/qmath/has043, URL https://doi.org/10.1093/qmath/has043 Martin [1990] Martin PP (1990) Representations of graph Temperley-Lieb algebras. Publ Res Inst Math Sci 26(3):485–503. 10.2977/prims/1195170958, URL https://doi.org/10.2977/prims/1195170958 Martin [2008] Martin S (2008) Schur algebras and representation theory, Cambridge Tracts in Mathematics, vol 112. Cambridge University Press, Cambridge, reprint of the 1993 original Munn [1955] Munn WD (1955) On semigroup algebras. Proc Cambridge Philos Soc 51:1–15 Munn [1957a] Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Martin P, Mazorchuk V (2014) On the representation theory of partial Brauer algebras. Q J Math 65(1):225–247. 10.1093/qmath/has043, URL https://doi.org/10.1093/qmath/has043 Martin [1990] Martin PP (1990) Representations of graph Temperley-Lieb algebras. Publ Res Inst Math Sci 26(3):485–503. 10.2977/prims/1195170958, URL https://doi.org/10.2977/prims/1195170958 Martin [2008] Martin S (2008) Schur algebras and representation theory, Cambridge Tracts in Mathematics, vol 112. Cambridge University Press, Cambridge, reprint of the 1993 original Munn [1955] Munn WD (1955) On semigroup algebras. Proc Cambridge Philos Soc 51:1–15 Munn [1957a] Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Martin PP (1990) Representations of graph Temperley-Lieb algebras. Publ Res Inst Math Sci 26(3):485–503. 10.2977/prims/1195170958, URL https://doi.org/10.2977/prims/1195170958 Martin [2008] Martin S (2008) Schur algebras and representation theory, Cambridge Tracts in Mathematics, vol 112. Cambridge University Press, Cambridge, reprint of the 1993 original Munn [1955] Munn WD (1955) On semigroup algebras. Proc Cambridge Philos Soc 51:1–15 Munn [1957a] Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Martin S (2008) Schur algebras and representation theory, Cambridge Tracts in Mathematics, vol 112. Cambridge University Press, Cambridge, reprint of the 1993 original Munn [1955] Munn WD (1955) On semigroup algebras. Proc Cambridge Philos Soc 51:1–15 Munn [1957a] Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Munn WD (1955) On semigroup algebras. Proc Cambridge Philos Soc 51:1–15 Munn [1957a] Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. 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J Combin Theory Ser A 113(2):325–351. 10.1016/j.jcta.2005.03.006, URL https://doi.org/10.1016/j.jcta.2005.03.006 Halverson [1996] Halverson T (1996) Characters of the centralizer algebras of mixed tensor representations of Gl(r,ℂ)Gl𝑟ℂ{\rm Gl}(r,\mathbb{C})roman_Gl ( italic_r , blackboard_C ) and the quantum group 𝔘q(ℊl(r,ℂ))subscript𝔘𝑞ℊ𝑙𝑟ℂ\mathfrak{U}_{q}({\mathcal{g}l}(r,\mathbb{C}))fraktur_U start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ( caligraphic_g italic_l ( italic_r , blackboard_C ) ). Pacific J Math 174(2):359–410. URL http://projecteuclid.org/euclid.pjm/1102365176 Halverson [2004] Halverson T (2004) Representations of the q𝑞qitalic_q-rook monoid. J Algebra 273(1):227–251. 10.1016/j.jalgebra.2003.11.002, URL http://dx.doi.org/10.1016/j.jalgebra.2003.11.002 Halverson and delMas [2014] Halverson T, delMas E (2014) Representations of the Rook-Brauer algebra. 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Inc., River Edge, NJ, 10.1142/9789812816689, URL http://dx.doi.org/10.1142/9789812816689, the theory of partial symmetries Martin [1994] Martin P (1994) Temperley-Lieb algebras for nonplanar statistical mechanics—the partition algebra construction. J Knot Theory Ramifications 3(1):51–82. 10.1142/S0218216594000071, URL http://dx.doi.org/10.1142/S0218216594000071 Martin [1996] Martin P (1996) The structure of the partition algebras. J Algebra 183(2):319–358. 10.1006/jabr.1996.0223, URL http://dx.doi.org/10.1006/jabr.1996.0223 Martin and Mazorchuk [2014] Martin P, Mazorchuk V (2014) On the representation theory of partial Brauer algebras. Q J Math 65(1):225–247. 10.1093/qmath/has043, URL https://doi.org/10.1093/qmath/has043 Martin [1990] Martin PP (1990) Representations of graph Temperley-Lieb algebras. Publ Res Inst Math Sci 26(3):485–503. 10.2977/prims/1195170958, URL https://doi.org/10.2977/prims/1195170958 Martin [2008] Martin S (2008) Schur algebras and representation theory, Cambridge Tracts in Mathematics, vol 112. Cambridge University Press, Cambridge, reprint of the 1993 original Munn [1955] Munn WD (1955) On semigroup algebras. Proc Cambridge Philos Soc 51:1–15 Munn [1957a] Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Graham JJ, Lehrer GI (1996) Cellular algebras. Invent Math 123(1):1–34. 10.1007/BF01232365, URL http://dx.doi.org/10.1007/BF01232365 Green [1980] Green JA (1980) Polynomial representations of GLnsubscriptGL𝑛{\rm GL}_{n}roman_GL start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT, Lecture Notes in Mathematics, vol 830. Springer-Verlag, Berlin Grood [2006] Grood C (2006) The rook partition algebra. J Combin Theory Ser A 113(2):325–351. 10.1016/j.jcta.2005.03.006, URL https://doi.org/10.1016/j.jcta.2005.03.006 Halverson [1996] Halverson T (1996) Characters of the centralizer algebras of mixed tensor representations of Gl(r,ℂ)Gl𝑟ℂ{\rm Gl}(r,\mathbb{C})roman_Gl ( italic_r , blackboard_C ) and the quantum group 𝔘q(ℊl(r,ℂ))subscript𝔘𝑞ℊ𝑙𝑟ℂ\mathfrak{U}_{q}({\mathcal{g}l}(r,\mathbb{C}))fraktur_U start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ( caligraphic_g italic_l ( italic_r , blackboard_C ) ). Pacific J Math 174(2):359–410. URL http://projecteuclid.org/euclid.pjm/1102365176 Halverson [2004] Halverson T (2004) Representations of the q𝑞qitalic_q-rook monoid. J Algebra 273(1):227–251. 10.1016/j.jalgebra.2003.11.002, URL http://dx.doi.org/10.1016/j.jalgebra.2003.11.002 Halverson and delMas [2014] Halverson T, delMas E (2014) Representations of the Rook-Brauer algebra. Comm Algebra 42(1):423–443. 10.1080/00927872.2012.716120, URL https://doi.org/10.1080/00927872.2012.716120 Halverson and Ram [2005] Halverson T, Ram A (2005) Partition algebras. European J Combin 26(6):869–921. 10.1016/j.ejc.2004.06.005, URL http://dx.doi.org/10.1016/j.ejc.2004.06.005 Jacobson [1989] Jacobson N (1989) Basic algebra. II, 2nd edn. W. H. Freeman and Company, New York Jones [1994] Jones VFR (1994) The Potts model and the symmetric group. In: Subfactors (Kyuzeso, 1993). World Sci. Publ., River Edge, NJ, p 259–267 Kudryavtseva and Mazorchuk [2008] Kudryavtseva G, Mazorchuk V (2008) Schur-Weyl dualities for symmetric inverse semigroups. J Pure Appl Algebra 212(8):1987–1995. 10.1016/j.jpaa.2007.12.004, URL https://doi.org/10.1016/j.jpaa.2007.12.004 Lawson [1998] Lawson MV (1998) Inverse semigroups. World Scientific Publishing Co. Inc., River Edge, NJ, 10.1142/9789812816689, URL http://dx.doi.org/10.1142/9789812816689, the theory of partial symmetries Martin [1994] Martin P (1994) Temperley-Lieb algebras for nonplanar statistical mechanics—the partition algebra construction. J Knot Theory Ramifications 3(1):51–82. 10.1142/S0218216594000071, URL http://dx.doi.org/10.1142/S0218216594000071 Martin [1996] Martin P (1996) The structure of the partition algebras. J Algebra 183(2):319–358. 10.1006/jabr.1996.0223, URL http://dx.doi.org/10.1006/jabr.1996.0223 Martin and Mazorchuk [2014] Martin P, Mazorchuk V (2014) On the representation theory of partial Brauer algebras. Q J Math 65(1):225–247. 10.1093/qmath/has043, URL https://doi.org/10.1093/qmath/has043 Martin [1990] Martin PP (1990) Representations of graph Temperley-Lieb algebras. Publ Res Inst Math Sci 26(3):485–503. 10.2977/prims/1195170958, URL https://doi.org/10.2977/prims/1195170958 Martin [2008] Martin S (2008) Schur algebras and representation theory, Cambridge Tracts in Mathematics, vol 112. Cambridge University Press, Cambridge, reprint of the 1993 original Munn [1955] Munn WD (1955) On semigroup algebras. Proc Cambridge Philos Soc 51:1–15 Munn [1957a] Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Green JA (1980) Polynomial representations of GLnsubscriptGL𝑛{\rm GL}_{n}roman_GL start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT, Lecture Notes in Mathematics, vol 830. Springer-Verlag, Berlin Grood [2006] Grood C (2006) The rook partition algebra. J Combin Theory Ser A 113(2):325–351. 10.1016/j.jcta.2005.03.006, URL https://doi.org/10.1016/j.jcta.2005.03.006 Halverson [1996] Halverson T (1996) Characters of the centralizer algebras of mixed tensor representations of Gl(r,ℂ)Gl𝑟ℂ{\rm Gl}(r,\mathbb{C})roman_Gl ( italic_r , blackboard_C ) and the quantum group 𝔘q(ℊl(r,ℂ))subscript𝔘𝑞ℊ𝑙𝑟ℂ\mathfrak{U}_{q}({\mathcal{g}l}(r,\mathbb{C}))fraktur_U start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ( caligraphic_g italic_l ( italic_r , blackboard_C ) ). Pacific J Math 174(2):359–410. URL http://projecteuclid.org/euclid.pjm/1102365176 Halverson [2004] Halverson T (2004) Representations of the q𝑞qitalic_q-rook monoid. J Algebra 273(1):227–251. 10.1016/j.jalgebra.2003.11.002, URL http://dx.doi.org/10.1016/j.jalgebra.2003.11.002 Halverson and delMas [2014] Halverson T, delMas E (2014) Representations of the Rook-Brauer algebra. Comm Algebra 42(1):423–443. 10.1080/00927872.2012.716120, URL https://doi.org/10.1080/00927872.2012.716120 Halverson and Ram [2005] Halverson T, Ram A (2005) Partition algebras. European J Combin 26(6):869–921. 10.1016/j.ejc.2004.06.005, URL http://dx.doi.org/10.1016/j.ejc.2004.06.005 Jacobson [1989] Jacobson N (1989) Basic algebra. II, 2nd edn. W. H. Freeman and Company, New York Jones [1994] Jones VFR (1994) The Potts model and the symmetric group. In: Subfactors (Kyuzeso, 1993). World Sci. Publ., River Edge, NJ, p 259–267 Kudryavtseva and Mazorchuk [2008] Kudryavtseva G, Mazorchuk V (2008) Schur-Weyl dualities for symmetric inverse semigroups. J Pure Appl Algebra 212(8):1987–1995. 10.1016/j.jpaa.2007.12.004, URL https://doi.org/10.1016/j.jpaa.2007.12.004 Lawson [1998] Lawson MV (1998) Inverse semigroups. World Scientific Publishing Co. Inc., River Edge, NJ, 10.1142/9789812816689, URL http://dx.doi.org/10.1142/9789812816689, the theory of partial symmetries Martin [1994] Martin P (1994) Temperley-Lieb algebras for nonplanar statistical mechanics—the partition algebra construction. J Knot Theory Ramifications 3(1):51–82. 10.1142/S0218216594000071, URL http://dx.doi.org/10.1142/S0218216594000071 Martin [1996] Martin P (1996) The structure of the partition algebras. J Algebra 183(2):319–358. 10.1006/jabr.1996.0223, URL http://dx.doi.org/10.1006/jabr.1996.0223 Martin and Mazorchuk [2014] Martin P, Mazorchuk V (2014) On the representation theory of partial Brauer algebras. Q J Math 65(1):225–247. 10.1093/qmath/has043, URL https://doi.org/10.1093/qmath/has043 Martin [1990] Martin PP (1990) Representations of graph Temperley-Lieb algebras. Publ Res Inst Math Sci 26(3):485–503. 10.2977/prims/1195170958, URL https://doi.org/10.2977/prims/1195170958 Martin [2008] Martin S (2008) Schur algebras and representation theory, Cambridge Tracts in Mathematics, vol 112. Cambridge University Press, Cambridge, reprint of the 1993 original Munn [1955] Munn WD (1955) On semigroup algebras. Proc Cambridge Philos Soc 51:1–15 Munn [1957a] Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Grood C (2006) The rook partition algebra. J Combin Theory Ser A 113(2):325–351. 10.1016/j.jcta.2005.03.006, URL https://doi.org/10.1016/j.jcta.2005.03.006 Halverson [1996] Halverson T (1996) Characters of the centralizer algebras of mixed tensor representations of Gl(r,ℂ)Gl𝑟ℂ{\rm Gl}(r,\mathbb{C})roman_Gl ( italic_r , blackboard_C ) and the quantum group 𝔘q(ℊl(r,ℂ))subscript𝔘𝑞ℊ𝑙𝑟ℂ\mathfrak{U}_{q}({\mathcal{g}l}(r,\mathbb{C}))fraktur_U start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ( caligraphic_g italic_l ( italic_r , blackboard_C ) ). Pacific J Math 174(2):359–410. URL http://projecteuclid.org/euclid.pjm/1102365176 Halverson [2004] Halverson T (2004) Representations of the q𝑞qitalic_q-rook monoid. J Algebra 273(1):227–251. 10.1016/j.jalgebra.2003.11.002, URL http://dx.doi.org/10.1016/j.jalgebra.2003.11.002 Halverson and delMas [2014] Halverson T, delMas E (2014) Representations of the Rook-Brauer algebra. Comm Algebra 42(1):423–443. 10.1080/00927872.2012.716120, URL https://doi.org/10.1080/00927872.2012.716120 Halverson and Ram [2005] Halverson T, Ram A (2005) Partition algebras. European J Combin 26(6):869–921. 10.1016/j.ejc.2004.06.005, URL http://dx.doi.org/10.1016/j.ejc.2004.06.005 Jacobson [1989] Jacobson N (1989) Basic algebra. II, 2nd edn. W. H. Freeman and Company, New York Jones [1994] Jones VFR (1994) The Potts model and the symmetric group. In: Subfactors (Kyuzeso, 1993). World Sci. Publ., River Edge, NJ, p 259–267 Kudryavtseva and Mazorchuk [2008] Kudryavtseva G, Mazorchuk V (2008) Schur-Weyl dualities for symmetric inverse semigroups. J Pure Appl Algebra 212(8):1987–1995. 10.1016/j.jpaa.2007.12.004, URL https://doi.org/10.1016/j.jpaa.2007.12.004 Lawson [1998] Lawson MV (1998) Inverse semigroups. World Scientific Publishing Co. Inc., River Edge, NJ, 10.1142/9789812816689, URL http://dx.doi.org/10.1142/9789812816689, the theory of partial symmetries Martin [1994] Martin P (1994) Temperley-Lieb algebras for nonplanar statistical mechanics—the partition algebra construction. J Knot Theory Ramifications 3(1):51–82. 10.1142/S0218216594000071, URL http://dx.doi.org/10.1142/S0218216594000071 Martin [1996] Martin P (1996) The structure of the partition algebras. J Algebra 183(2):319–358. 10.1006/jabr.1996.0223, URL http://dx.doi.org/10.1006/jabr.1996.0223 Martin and Mazorchuk [2014] Martin P, Mazorchuk V (2014) On the representation theory of partial Brauer algebras. Q J Math 65(1):225–247. 10.1093/qmath/has043, URL https://doi.org/10.1093/qmath/has043 Martin [1990] Martin PP (1990) Representations of graph Temperley-Lieb algebras. Publ Res Inst Math Sci 26(3):485–503. 10.2977/prims/1195170958, URL https://doi.org/10.2977/prims/1195170958 Martin [2008] Martin S (2008) Schur algebras and representation theory, Cambridge Tracts in Mathematics, vol 112. Cambridge University Press, Cambridge, reprint of the 1993 original Munn [1955] Munn WD (1955) On semigroup algebras. Proc Cambridge Philos Soc 51:1–15 Munn [1957a] Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Halverson T (1996) Characters of the centralizer algebras of mixed tensor representations of Gl(r,ℂ)Gl𝑟ℂ{\rm Gl}(r,\mathbb{C})roman_Gl ( italic_r , blackboard_C ) and the quantum group 𝔘q(ℊl(r,ℂ))subscript𝔘𝑞ℊ𝑙𝑟ℂ\mathfrak{U}_{q}({\mathcal{g}l}(r,\mathbb{C}))fraktur_U start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ( caligraphic_g italic_l ( italic_r , blackboard_C ) ). Pacific J Math 174(2):359–410. URL http://projecteuclid.org/euclid.pjm/1102365176 Halverson [2004] Halverson T (2004) Representations of the q𝑞qitalic_q-rook monoid. J Algebra 273(1):227–251. 10.1016/j.jalgebra.2003.11.002, URL http://dx.doi.org/10.1016/j.jalgebra.2003.11.002 Halverson and delMas [2014] Halverson T, delMas E (2014) Representations of the Rook-Brauer algebra. Comm Algebra 42(1):423–443. 10.1080/00927872.2012.716120, URL https://doi.org/10.1080/00927872.2012.716120 Halverson and Ram [2005] Halverson T, Ram A (2005) Partition algebras. European J Combin 26(6):869–921. 10.1016/j.ejc.2004.06.005, URL http://dx.doi.org/10.1016/j.ejc.2004.06.005 Jacobson [1989] Jacobson N (1989) Basic algebra. II, 2nd edn. W. H. Freeman and Company, New York Jones [1994] Jones VFR (1994) The Potts model and the symmetric group. In: Subfactors (Kyuzeso, 1993). World Sci. Publ., River Edge, NJ, p 259–267 Kudryavtseva and Mazorchuk [2008] Kudryavtseva G, Mazorchuk V (2008) Schur-Weyl dualities for symmetric inverse semigroups. J Pure Appl Algebra 212(8):1987–1995. 10.1016/j.jpaa.2007.12.004, URL https://doi.org/10.1016/j.jpaa.2007.12.004 Lawson [1998] Lawson MV (1998) Inverse semigroups. World Scientific Publishing Co. Inc., River Edge, NJ, 10.1142/9789812816689, URL http://dx.doi.org/10.1142/9789812816689, the theory of partial symmetries Martin [1994] Martin P (1994) Temperley-Lieb algebras for nonplanar statistical mechanics—the partition algebra construction. J Knot Theory Ramifications 3(1):51–82. 10.1142/S0218216594000071, URL http://dx.doi.org/10.1142/S0218216594000071 Martin [1996] Martin P (1996) The structure of the partition algebras. J Algebra 183(2):319–358. 10.1006/jabr.1996.0223, URL http://dx.doi.org/10.1006/jabr.1996.0223 Martin and Mazorchuk [2014] Martin P, Mazorchuk V (2014) On the representation theory of partial Brauer algebras. Q J Math 65(1):225–247. 10.1093/qmath/has043, URL https://doi.org/10.1093/qmath/has043 Martin [1990] Martin PP (1990) Representations of graph Temperley-Lieb algebras. Publ Res Inst Math Sci 26(3):485–503. 10.2977/prims/1195170958, URL https://doi.org/10.2977/prims/1195170958 Martin [2008] Martin S (2008) Schur algebras and representation theory, Cambridge Tracts in Mathematics, vol 112. Cambridge University Press, Cambridge, reprint of the 1993 original Munn [1955] Munn WD (1955) On semigroup algebras. Proc Cambridge Philos Soc 51:1–15 Munn [1957a] Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Halverson T (2004) Representations of the q𝑞qitalic_q-rook monoid. J Algebra 273(1):227–251. 10.1016/j.jalgebra.2003.11.002, URL http://dx.doi.org/10.1016/j.jalgebra.2003.11.002 Halverson and delMas [2014] Halverson T, delMas E (2014) Representations of the Rook-Brauer algebra. Comm Algebra 42(1):423–443. 10.1080/00927872.2012.716120, URL https://doi.org/10.1080/00927872.2012.716120 Halverson and Ram [2005] Halverson T, Ram A (2005) Partition algebras. European J Combin 26(6):869–921. 10.1016/j.ejc.2004.06.005, URL http://dx.doi.org/10.1016/j.ejc.2004.06.005 Jacobson [1989] Jacobson N (1989) Basic algebra. II, 2nd edn. W. H. Freeman and Company, New York Jones [1994] Jones VFR (1994) The Potts model and the symmetric group. In: Subfactors (Kyuzeso, 1993). World Sci. Publ., River Edge, NJ, p 259–267 Kudryavtseva and Mazorchuk [2008] Kudryavtseva G, Mazorchuk V (2008) Schur-Weyl dualities for symmetric inverse semigroups. J Pure Appl Algebra 212(8):1987–1995. 10.1016/j.jpaa.2007.12.004, URL https://doi.org/10.1016/j.jpaa.2007.12.004 Lawson [1998] Lawson MV (1998) Inverse semigroups. World Scientific Publishing Co. Inc., River Edge, NJ, 10.1142/9789812816689, URL http://dx.doi.org/10.1142/9789812816689, the theory of partial symmetries Martin [1994] Martin P (1994) Temperley-Lieb algebras for nonplanar statistical mechanics—the partition algebra construction. J Knot Theory Ramifications 3(1):51–82. 10.1142/S0218216594000071, URL http://dx.doi.org/10.1142/S0218216594000071 Martin [1996] Martin P (1996) The structure of the partition algebras. J Algebra 183(2):319–358. 10.1006/jabr.1996.0223, URL http://dx.doi.org/10.1006/jabr.1996.0223 Martin and Mazorchuk [2014] Martin P, Mazorchuk V (2014) On the representation theory of partial Brauer algebras. Q J Math 65(1):225–247. 10.1093/qmath/has043, URL https://doi.org/10.1093/qmath/has043 Martin [1990] Martin PP (1990) Representations of graph Temperley-Lieb algebras. Publ Res Inst Math Sci 26(3):485–503. 10.2977/prims/1195170958, URL https://doi.org/10.2977/prims/1195170958 Martin [2008] Martin S (2008) Schur algebras and representation theory, Cambridge Tracts in Mathematics, vol 112. Cambridge University Press, Cambridge, reprint of the 1993 original Munn [1955] Munn WD (1955) On semigroup algebras. Proc Cambridge Philos Soc 51:1–15 Munn [1957a] Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Halverson T, delMas E (2014) Representations of the Rook-Brauer algebra. Comm Algebra 42(1):423–443. 10.1080/00927872.2012.716120, URL https://doi.org/10.1080/00927872.2012.716120 Halverson and Ram [2005] Halverson T, Ram A (2005) Partition algebras. European J Combin 26(6):869–921. 10.1016/j.ejc.2004.06.005, URL http://dx.doi.org/10.1016/j.ejc.2004.06.005 Jacobson [1989] Jacobson N (1989) Basic algebra. II, 2nd edn. W. H. Freeman and Company, New York Jones [1994] Jones VFR (1994) The Potts model and the symmetric group. In: Subfactors (Kyuzeso, 1993). World Sci. Publ., River Edge, NJ, p 259–267 Kudryavtseva and Mazorchuk [2008] Kudryavtseva G, Mazorchuk V (2008) Schur-Weyl dualities for symmetric inverse semigroups. J Pure Appl Algebra 212(8):1987–1995. 10.1016/j.jpaa.2007.12.004, URL https://doi.org/10.1016/j.jpaa.2007.12.004 Lawson [1998] Lawson MV (1998) Inverse semigroups. World Scientific Publishing Co. Inc., River Edge, NJ, 10.1142/9789812816689, URL http://dx.doi.org/10.1142/9789812816689, the theory of partial symmetries Martin [1994] Martin P (1994) Temperley-Lieb algebras for nonplanar statistical mechanics—the partition algebra construction. J Knot Theory Ramifications 3(1):51–82. 10.1142/S0218216594000071, URL http://dx.doi.org/10.1142/S0218216594000071 Martin [1996] Martin P (1996) The structure of the partition algebras. J Algebra 183(2):319–358. 10.1006/jabr.1996.0223, URL http://dx.doi.org/10.1006/jabr.1996.0223 Martin and Mazorchuk [2014] Martin P, Mazorchuk V (2014) On the representation theory of partial Brauer algebras. Q J Math 65(1):225–247. 10.1093/qmath/has043, URL https://doi.org/10.1093/qmath/has043 Martin [1990] Martin PP (1990) Representations of graph Temperley-Lieb algebras. Publ Res Inst Math Sci 26(3):485–503. 10.2977/prims/1195170958, URL https://doi.org/10.2977/prims/1195170958 Martin [2008] Martin S (2008) Schur algebras and representation theory, Cambridge Tracts in Mathematics, vol 112. Cambridge University Press, Cambridge, reprint of the 1993 original Munn [1955] Munn WD (1955) On semigroup algebras. Proc Cambridge Philos Soc 51:1–15 Munn [1957a] Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Halverson T, Ram A (2005) Partition algebras. European J Combin 26(6):869–921. 10.1016/j.ejc.2004.06.005, URL http://dx.doi.org/10.1016/j.ejc.2004.06.005 Jacobson [1989] Jacobson N (1989) Basic algebra. II, 2nd edn. W. H. Freeman and Company, New York Jones [1994] Jones VFR (1994) The Potts model and the symmetric group. In: Subfactors (Kyuzeso, 1993). World Sci. Publ., River Edge, NJ, p 259–267 Kudryavtseva and Mazorchuk [2008] Kudryavtseva G, Mazorchuk V (2008) Schur-Weyl dualities for symmetric inverse semigroups. J Pure Appl Algebra 212(8):1987–1995. 10.1016/j.jpaa.2007.12.004, URL https://doi.org/10.1016/j.jpaa.2007.12.004 Lawson [1998] Lawson MV (1998) Inverse semigroups. World Scientific Publishing Co. Inc., River Edge, NJ, 10.1142/9789812816689, URL http://dx.doi.org/10.1142/9789812816689, the theory of partial symmetries Martin [1994] Martin P (1994) Temperley-Lieb algebras for nonplanar statistical mechanics—the partition algebra construction. J Knot Theory Ramifications 3(1):51–82. 10.1142/S0218216594000071, URL http://dx.doi.org/10.1142/S0218216594000071 Martin [1996] Martin P (1996) The structure of the partition algebras. J Algebra 183(2):319–358. 10.1006/jabr.1996.0223, URL http://dx.doi.org/10.1006/jabr.1996.0223 Martin and Mazorchuk [2014] Martin P, Mazorchuk V (2014) On the representation theory of partial Brauer algebras. Q J Math 65(1):225–247. 10.1093/qmath/has043, URL https://doi.org/10.1093/qmath/has043 Martin [1990] Martin PP (1990) Representations of graph Temperley-Lieb algebras. Publ Res Inst Math Sci 26(3):485–503. 10.2977/prims/1195170958, URL https://doi.org/10.2977/prims/1195170958 Martin [2008] Martin S (2008) Schur algebras and representation theory, Cambridge Tracts in Mathematics, vol 112. Cambridge University Press, Cambridge, reprint of the 1993 original Munn [1955] Munn WD (1955) On semigroup algebras. Proc Cambridge Philos Soc 51:1–15 Munn [1957a] Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Jacobson N (1989) Basic algebra. II, 2nd edn. W. H. Freeman and Company, New York Jones [1994] Jones VFR (1994) The Potts model and the symmetric group. In: Subfactors (Kyuzeso, 1993). World Sci. Publ., River Edge, NJ, p 259–267 Kudryavtseva and Mazorchuk [2008] Kudryavtseva G, Mazorchuk V (2008) Schur-Weyl dualities for symmetric inverse semigroups. J Pure Appl Algebra 212(8):1987–1995. 10.1016/j.jpaa.2007.12.004, URL https://doi.org/10.1016/j.jpaa.2007.12.004 Lawson [1998] Lawson MV (1998) Inverse semigroups. World Scientific Publishing Co. Inc., River Edge, NJ, 10.1142/9789812816689, URL http://dx.doi.org/10.1142/9789812816689, the theory of partial symmetries Martin [1994] Martin P (1994) Temperley-Lieb algebras for nonplanar statistical mechanics—the partition algebra construction. J Knot Theory Ramifications 3(1):51–82. 10.1142/S0218216594000071, URL http://dx.doi.org/10.1142/S0218216594000071 Martin [1996] Martin P (1996) The structure of the partition algebras. J Algebra 183(2):319–358. 10.1006/jabr.1996.0223, URL http://dx.doi.org/10.1006/jabr.1996.0223 Martin and Mazorchuk [2014] Martin P, Mazorchuk V (2014) On the representation theory of partial Brauer algebras. Q J Math 65(1):225–247. 10.1093/qmath/has043, URL https://doi.org/10.1093/qmath/has043 Martin [1990] Martin PP (1990) Representations of graph Temperley-Lieb algebras. Publ Res Inst Math Sci 26(3):485–503. 10.2977/prims/1195170958, URL https://doi.org/10.2977/prims/1195170958 Martin [2008] Martin S (2008) Schur algebras and representation theory, Cambridge Tracts in Mathematics, vol 112. Cambridge University Press, Cambridge, reprint of the 1993 original Munn [1955] Munn WD (1955) On semigroup algebras. Proc Cambridge Philos Soc 51:1–15 Munn [1957a] Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Jones VFR (1994) The Potts model and the symmetric group. In: Subfactors (Kyuzeso, 1993). World Sci. Publ., River Edge, NJ, p 259–267 Kudryavtseva and Mazorchuk [2008] Kudryavtseva G, Mazorchuk V (2008) Schur-Weyl dualities for symmetric inverse semigroups. J Pure Appl Algebra 212(8):1987–1995. 10.1016/j.jpaa.2007.12.004, URL https://doi.org/10.1016/j.jpaa.2007.12.004 Lawson [1998] Lawson MV (1998) Inverse semigroups. World Scientific Publishing Co. Inc., River Edge, NJ, 10.1142/9789812816689, URL http://dx.doi.org/10.1142/9789812816689, the theory of partial symmetries Martin [1994] Martin P (1994) Temperley-Lieb algebras for nonplanar statistical mechanics—the partition algebra construction. J Knot Theory Ramifications 3(1):51–82. 10.1142/S0218216594000071, URL http://dx.doi.org/10.1142/S0218216594000071 Martin [1996] Martin P (1996) The structure of the partition algebras. J Algebra 183(2):319–358. 10.1006/jabr.1996.0223, URL http://dx.doi.org/10.1006/jabr.1996.0223 Martin and Mazorchuk [2014] Martin P, Mazorchuk V (2014) On the representation theory of partial Brauer algebras. Q J Math 65(1):225–247. 10.1093/qmath/has043, URL https://doi.org/10.1093/qmath/has043 Martin [1990] Martin PP (1990) Representations of graph Temperley-Lieb algebras. Publ Res Inst Math Sci 26(3):485–503. 10.2977/prims/1195170958, URL https://doi.org/10.2977/prims/1195170958 Martin [2008] Martin S (2008) Schur algebras and representation theory, Cambridge Tracts in Mathematics, vol 112. Cambridge University Press, Cambridge, reprint of the 1993 original Munn [1955] Munn WD (1955) On semigroup algebras. Proc Cambridge Philos Soc 51:1–15 Munn [1957a] Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Kudryavtseva G, Mazorchuk V (2008) Schur-Weyl dualities for symmetric inverse semigroups. J Pure Appl Algebra 212(8):1987–1995. 10.1016/j.jpaa.2007.12.004, URL https://doi.org/10.1016/j.jpaa.2007.12.004 Lawson [1998] Lawson MV (1998) Inverse semigroups. World Scientific Publishing Co. Inc., River Edge, NJ, 10.1142/9789812816689, URL http://dx.doi.org/10.1142/9789812816689, the theory of partial symmetries Martin [1994] Martin P (1994) Temperley-Lieb algebras for nonplanar statistical mechanics—the partition algebra construction. J Knot Theory Ramifications 3(1):51–82. 10.1142/S0218216594000071, URL http://dx.doi.org/10.1142/S0218216594000071 Martin [1996] Martin P (1996) The structure of the partition algebras. J Algebra 183(2):319–358. 10.1006/jabr.1996.0223, URL http://dx.doi.org/10.1006/jabr.1996.0223 Martin and Mazorchuk [2014] Martin P, Mazorchuk V (2014) On the representation theory of partial Brauer algebras. Q J Math 65(1):225–247. 10.1093/qmath/has043, URL https://doi.org/10.1093/qmath/has043 Martin [1990] Martin PP (1990) Representations of graph Temperley-Lieb algebras. Publ Res Inst Math Sci 26(3):485–503. 10.2977/prims/1195170958, URL https://doi.org/10.2977/prims/1195170958 Martin [2008] Martin S (2008) Schur algebras and representation theory, Cambridge Tracts in Mathematics, vol 112. Cambridge University Press, Cambridge, reprint of the 1993 original Munn [1955] Munn WD (1955) On semigroup algebras. Proc Cambridge Philos Soc 51:1–15 Munn [1957a] Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Lawson MV (1998) Inverse semigroups. World Scientific Publishing Co. Inc., River Edge, NJ, 10.1142/9789812816689, URL http://dx.doi.org/10.1142/9789812816689, the theory of partial symmetries Martin [1994] Martin P (1994) Temperley-Lieb algebras for nonplanar statistical mechanics—the partition algebra construction. J Knot Theory Ramifications 3(1):51–82. 10.1142/S0218216594000071, URL http://dx.doi.org/10.1142/S0218216594000071 Martin [1996] Martin P (1996) The structure of the partition algebras. J Algebra 183(2):319–358. 10.1006/jabr.1996.0223, URL http://dx.doi.org/10.1006/jabr.1996.0223 Martin and Mazorchuk [2014] Martin P, Mazorchuk V (2014) On the representation theory of partial Brauer algebras. Q J Math 65(1):225–247. 10.1093/qmath/has043, URL https://doi.org/10.1093/qmath/has043 Martin [1990] Martin PP (1990) Representations of graph Temperley-Lieb algebras. Publ Res Inst Math Sci 26(3):485–503. 10.2977/prims/1195170958, URL https://doi.org/10.2977/prims/1195170958 Martin [2008] Martin S (2008) Schur algebras and representation theory, Cambridge Tracts in Mathematics, vol 112. Cambridge University Press, Cambridge, reprint of the 1993 original Munn [1955] Munn WD (1955) On semigroup algebras. Proc Cambridge Philos Soc 51:1–15 Munn [1957a] Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Martin P (1994) Temperley-Lieb algebras for nonplanar statistical mechanics—the partition algebra construction. J Knot Theory Ramifications 3(1):51–82. 10.1142/S0218216594000071, URL http://dx.doi.org/10.1142/S0218216594000071 Martin [1996] Martin P (1996) The structure of the partition algebras. J Algebra 183(2):319–358. 10.1006/jabr.1996.0223, URL http://dx.doi.org/10.1006/jabr.1996.0223 Martin and Mazorchuk [2014] Martin P, Mazorchuk V (2014) On the representation theory of partial Brauer algebras. Q J Math 65(1):225–247. 10.1093/qmath/has043, URL https://doi.org/10.1093/qmath/has043 Martin [1990] Martin PP (1990) Representations of graph Temperley-Lieb algebras. Publ Res Inst Math Sci 26(3):485–503. 10.2977/prims/1195170958, URL https://doi.org/10.2977/prims/1195170958 Martin [2008] Martin S (2008) Schur algebras and representation theory, Cambridge Tracts in Mathematics, vol 112. Cambridge University Press, Cambridge, reprint of the 1993 original Munn [1955] Munn WD (1955) On semigroup algebras. Proc Cambridge Philos Soc 51:1–15 Munn [1957a] Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Martin P (1996) The structure of the partition algebras. J Algebra 183(2):319–358. 10.1006/jabr.1996.0223, URL http://dx.doi.org/10.1006/jabr.1996.0223 Martin and Mazorchuk [2014] Martin P, Mazorchuk V (2014) On the representation theory of partial Brauer algebras. Q J Math 65(1):225–247. 10.1093/qmath/has043, URL https://doi.org/10.1093/qmath/has043 Martin [1990] Martin PP (1990) Representations of graph Temperley-Lieb algebras. Publ Res Inst Math Sci 26(3):485–503. 10.2977/prims/1195170958, URL https://doi.org/10.2977/prims/1195170958 Martin [2008] Martin S (2008) Schur algebras and representation theory, Cambridge Tracts in Mathematics, vol 112. Cambridge University Press, Cambridge, reprint of the 1993 original Munn [1955] Munn WD (1955) On semigroup algebras. Proc Cambridge Philos Soc 51:1–15 Munn [1957a] Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Martin P, Mazorchuk V (2014) On the representation theory of partial Brauer algebras. Q J Math 65(1):225–247. 10.1093/qmath/has043, URL https://doi.org/10.1093/qmath/has043 Martin [1990] Martin PP (1990) Representations of graph Temperley-Lieb algebras. Publ Res Inst Math Sci 26(3):485–503. 10.2977/prims/1195170958, URL https://doi.org/10.2977/prims/1195170958 Martin [2008] Martin S (2008) Schur algebras and representation theory, Cambridge Tracts in Mathematics, vol 112. Cambridge University Press, Cambridge, reprint of the 1993 original Munn [1955] Munn WD (1955) On semigroup algebras. Proc Cambridge Philos Soc 51:1–15 Munn [1957a] Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Martin PP (1990) Representations of graph Temperley-Lieb algebras. Publ Res Inst Math Sci 26(3):485–503. 10.2977/prims/1195170958, URL https://doi.org/10.2977/prims/1195170958 Martin [2008] Martin S (2008) Schur algebras and representation theory, Cambridge Tracts in Mathematics, vol 112. Cambridge University Press, Cambridge, reprint of the 1993 original Munn [1955] Munn WD (1955) On semigroup algebras. Proc Cambridge Philos Soc 51:1–15 Munn [1957a] Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Martin S (2008) Schur algebras and representation theory, Cambridge Tracts in Mathematics, vol 112. Cambridge University Press, Cambridge, reprint of the 1993 original Munn [1955] Munn WD (1955) On semigroup algebras. Proc Cambridge Philos Soc 51:1–15 Munn [1957a] Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Munn WD (1955) On semigroup algebras. Proc Cambridge Philos Soc 51:1–15 Munn [1957a] Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J.
- Graham JJ, Lehrer GI (1996) Cellular algebras. Invent Math 123(1):1–34. 10.1007/BF01232365, URL http://dx.doi.org/10.1007/BF01232365 Green [1980] Green JA (1980) Polynomial representations of GLnsubscriptGL𝑛{\rm GL}_{n}roman_GL start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT, Lecture Notes in Mathematics, vol 830. Springer-Verlag, Berlin Grood [2006] Grood C (2006) The rook partition algebra. J Combin Theory Ser A 113(2):325–351. 10.1016/j.jcta.2005.03.006, URL https://doi.org/10.1016/j.jcta.2005.03.006 Halverson [1996] Halverson T (1996) Characters of the centralizer algebras of mixed tensor representations of Gl(r,ℂ)Gl𝑟ℂ{\rm Gl}(r,\mathbb{C})roman_Gl ( italic_r , blackboard_C ) and the quantum group 𝔘q(ℊl(r,ℂ))subscript𝔘𝑞ℊ𝑙𝑟ℂ\mathfrak{U}_{q}({\mathcal{g}l}(r,\mathbb{C}))fraktur_U start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ( caligraphic_g italic_l ( italic_r , blackboard_C ) ). Pacific J Math 174(2):359–410. URL http://projecteuclid.org/euclid.pjm/1102365176 Halverson [2004] Halverson T (2004) Representations of the q𝑞qitalic_q-rook monoid. J Algebra 273(1):227–251. 10.1016/j.jalgebra.2003.11.002, URL http://dx.doi.org/10.1016/j.jalgebra.2003.11.002 Halverson and delMas [2014] Halverson T, delMas E (2014) Representations of the Rook-Brauer algebra. Comm Algebra 42(1):423–443. 10.1080/00927872.2012.716120, URL https://doi.org/10.1080/00927872.2012.716120 Halverson and Ram [2005] Halverson T, Ram A (2005) Partition algebras. European J Combin 26(6):869–921. 10.1016/j.ejc.2004.06.005, URL http://dx.doi.org/10.1016/j.ejc.2004.06.005 Jacobson [1989] Jacobson N (1989) Basic algebra. II, 2nd edn. W. H. Freeman and Company, New York Jones [1994] Jones VFR (1994) The Potts model and the symmetric group. In: Subfactors (Kyuzeso, 1993). World Sci. Publ., River Edge, NJ, p 259–267 Kudryavtseva and Mazorchuk [2008] Kudryavtseva G, Mazorchuk V (2008) Schur-Weyl dualities for symmetric inverse semigroups. J Pure Appl Algebra 212(8):1987–1995. 10.1016/j.jpaa.2007.12.004, URL https://doi.org/10.1016/j.jpaa.2007.12.004 Lawson [1998] Lawson MV (1998) Inverse semigroups. World Scientific Publishing Co. Inc., River Edge, NJ, 10.1142/9789812816689, URL http://dx.doi.org/10.1142/9789812816689, the theory of partial symmetries Martin [1994] Martin P (1994) Temperley-Lieb algebras for nonplanar statistical mechanics—the partition algebra construction. J Knot Theory Ramifications 3(1):51–82. 10.1142/S0218216594000071, URL http://dx.doi.org/10.1142/S0218216594000071 Martin [1996] Martin P (1996) The structure of the partition algebras. J Algebra 183(2):319–358. 10.1006/jabr.1996.0223, URL http://dx.doi.org/10.1006/jabr.1996.0223 Martin and Mazorchuk [2014] Martin P, Mazorchuk V (2014) On the representation theory of partial Brauer algebras. Q J Math 65(1):225–247. 10.1093/qmath/has043, URL https://doi.org/10.1093/qmath/has043 Martin [1990] Martin PP (1990) Representations of graph Temperley-Lieb algebras. Publ Res Inst Math Sci 26(3):485–503. 10.2977/prims/1195170958, URL https://doi.org/10.2977/prims/1195170958 Martin [2008] Martin S (2008) Schur algebras and representation theory, Cambridge Tracts in Mathematics, vol 112. Cambridge University Press, Cambridge, reprint of the 1993 original Munn [1955] Munn WD (1955) On semigroup algebras. Proc Cambridge Philos Soc 51:1–15 Munn [1957a] Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Green JA (1980) Polynomial representations of GLnsubscriptGL𝑛{\rm GL}_{n}roman_GL start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT, Lecture Notes in Mathematics, vol 830. Springer-Verlag, Berlin Grood [2006] Grood C (2006) The rook partition algebra. J Combin Theory Ser A 113(2):325–351. 10.1016/j.jcta.2005.03.006, URL https://doi.org/10.1016/j.jcta.2005.03.006 Halverson [1996] Halverson T (1996) Characters of the centralizer algebras of mixed tensor representations of Gl(r,ℂ)Gl𝑟ℂ{\rm Gl}(r,\mathbb{C})roman_Gl ( italic_r , blackboard_C ) and the quantum group 𝔘q(ℊl(r,ℂ))subscript𝔘𝑞ℊ𝑙𝑟ℂ\mathfrak{U}_{q}({\mathcal{g}l}(r,\mathbb{C}))fraktur_U start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ( caligraphic_g italic_l ( italic_r , blackboard_C ) ). Pacific J Math 174(2):359–410. URL http://projecteuclid.org/euclid.pjm/1102365176 Halverson [2004] Halverson T (2004) Representations of the q𝑞qitalic_q-rook monoid. J Algebra 273(1):227–251. 10.1016/j.jalgebra.2003.11.002, URL http://dx.doi.org/10.1016/j.jalgebra.2003.11.002 Halverson and delMas [2014] Halverson T, delMas E (2014) Representations of the Rook-Brauer algebra. Comm Algebra 42(1):423–443. 10.1080/00927872.2012.716120, URL https://doi.org/10.1080/00927872.2012.716120 Halverson and Ram [2005] Halverson T, Ram A (2005) Partition algebras. European J Combin 26(6):869–921. 10.1016/j.ejc.2004.06.005, URL http://dx.doi.org/10.1016/j.ejc.2004.06.005 Jacobson [1989] Jacobson N (1989) Basic algebra. II, 2nd edn. W. H. Freeman and Company, New York Jones [1994] Jones VFR (1994) The Potts model and the symmetric group. In: Subfactors (Kyuzeso, 1993). World Sci. Publ., River Edge, NJ, p 259–267 Kudryavtseva and Mazorchuk [2008] Kudryavtseva G, Mazorchuk V (2008) Schur-Weyl dualities for symmetric inverse semigroups. J Pure Appl Algebra 212(8):1987–1995. 10.1016/j.jpaa.2007.12.004, URL https://doi.org/10.1016/j.jpaa.2007.12.004 Lawson [1998] Lawson MV (1998) Inverse semigroups. World Scientific Publishing Co. Inc., River Edge, NJ, 10.1142/9789812816689, URL http://dx.doi.org/10.1142/9789812816689, the theory of partial symmetries Martin [1994] Martin P (1994) Temperley-Lieb algebras for nonplanar statistical mechanics—the partition algebra construction. J Knot Theory Ramifications 3(1):51–82. 10.1142/S0218216594000071, URL http://dx.doi.org/10.1142/S0218216594000071 Martin [1996] Martin P (1996) The structure of the partition algebras. J Algebra 183(2):319–358. 10.1006/jabr.1996.0223, URL http://dx.doi.org/10.1006/jabr.1996.0223 Martin and Mazorchuk [2014] Martin P, Mazorchuk V (2014) On the representation theory of partial Brauer algebras. Q J Math 65(1):225–247. 10.1093/qmath/has043, URL https://doi.org/10.1093/qmath/has043 Martin [1990] Martin PP (1990) Representations of graph Temperley-Lieb algebras. Publ Res Inst Math Sci 26(3):485–503. 10.2977/prims/1195170958, URL https://doi.org/10.2977/prims/1195170958 Martin [2008] Martin S (2008) Schur algebras and representation theory, Cambridge Tracts in Mathematics, vol 112. Cambridge University Press, Cambridge, reprint of the 1993 original Munn [1955] Munn WD (1955) On semigroup algebras. Proc Cambridge Philos Soc 51:1–15 Munn [1957a] Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Grood C (2006) The rook partition algebra. J Combin Theory Ser A 113(2):325–351. 10.1016/j.jcta.2005.03.006, URL https://doi.org/10.1016/j.jcta.2005.03.006 Halverson [1996] Halverson T (1996) Characters of the centralizer algebras of mixed tensor representations of Gl(r,ℂ)Gl𝑟ℂ{\rm Gl}(r,\mathbb{C})roman_Gl ( italic_r , blackboard_C ) and the quantum group 𝔘q(ℊl(r,ℂ))subscript𝔘𝑞ℊ𝑙𝑟ℂ\mathfrak{U}_{q}({\mathcal{g}l}(r,\mathbb{C}))fraktur_U start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ( caligraphic_g italic_l ( italic_r , blackboard_C ) ). Pacific J Math 174(2):359–410. URL http://projecteuclid.org/euclid.pjm/1102365176 Halverson [2004] Halverson T (2004) Representations of the q𝑞qitalic_q-rook monoid. J Algebra 273(1):227–251. 10.1016/j.jalgebra.2003.11.002, URL http://dx.doi.org/10.1016/j.jalgebra.2003.11.002 Halverson and delMas [2014] Halverson T, delMas E (2014) Representations of the Rook-Brauer algebra. Comm Algebra 42(1):423–443. 10.1080/00927872.2012.716120, URL https://doi.org/10.1080/00927872.2012.716120 Halverson and Ram [2005] Halverson T, Ram A (2005) Partition algebras. European J Combin 26(6):869–921. 10.1016/j.ejc.2004.06.005, URL http://dx.doi.org/10.1016/j.ejc.2004.06.005 Jacobson [1989] Jacobson N (1989) Basic algebra. II, 2nd edn. W. H. Freeman and Company, New York Jones [1994] Jones VFR (1994) The Potts model and the symmetric group. In: Subfactors (Kyuzeso, 1993). World Sci. Publ., River Edge, NJ, p 259–267 Kudryavtseva and Mazorchuk [2008] Kudryavtseva G, Mazorchuk V (2008) Schur-Weyl dualities for symmetric inverse semigroups. J Pure Appl Algebra 212(8):1987–1995. 10.1016/j.jpaa.2007.12.004, URL https://doi.org/10.1016/j.jpaa.2007.12.004 Lawson [1998] Lawson MV (1998) Inverse semigroups. World Scientific Publishing Co. Inc., River Edge, NJ, 10.1142/9789812816689, URL http://dx.doi.org/10.1142/9789812816689, the theory of partial symmetries Martin [1994] Martin P (1994) Temperley-Lieb algebras for nonplanar statistical mechanics—the partition algebra construction. J Knot Theory Ramifications 3(1):51–82. 10.1142/S0218216594000071, URL http://dx.doi.org/10.1142/S0218216594000071 Martin [1996] Martin P (1996) The structure of the partition algebras. J Algebra 183(2):319–358. 10.1006/jabr.1996.0223, URL http://dx.doi.org/10.1006/jabr.1996.0223 Martin and Mazorchuk [2014] Martin P, Mazorchuk V (2014) On the representation theory of partial Brauer algebras. Q J Math 65(1):225–247. 10.1093/qmath/has043, URL https://doi.org/10.1093/qmath/has043 Martin [1990] Martin PP (1990) Representations of graph Temperley-Lieb algebras. Publ Res Inst Math Sci 26(3):485–503. 10.2977/prims/1195170958, URL https://doi.org/10.2977/prims/1195170958 Martin [2008] Martin S (2008) Schur algebras and representation theory, Cambridge Tracts in Mathematics, vol 112. Cambridge University Press, Cambridge, reprint of the 1993 original Munn [1955] Munn WD (1955) On semigroup algebras. Proc Cambridge Philos Soc 51:1–15 Munn [1957a] Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Halverson T (1996) Characters of the centralizer algebras of mixed tensor representations of Gl(r,ℂ)Gl𝑟ℂ{\rm Gl}(r,\mathbb{C})roman_Gl ( italic_r , blackboard_C ) and the quantum group 𝔘q(ℊl(r,ℂ))subscript𝔘𝑞ℊ𝑙𝑟ℂ\mathfrak{U}_{q}({\mathcal{g}l}(r,\mathbb{C}))fraktur_U start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ( caligraphic_g italic_l ( italic_r , blackboard_C ) ). Pacific J Math 174(2):359–410. URL http://projecteuclid.org/euclid.pjm/1102365176 Halverson [2004] Halverson T (2004) Representations of the q𝑞qitalic_q-rook monoid. J Algebra 273(1):227–251. 10.1016/j.jalgebra.2003.11.002, URL http://dx.doi.org/10.1016/j.jalgebra.2003.11.002 Halverson and delMas [2014] Halverson T, delMas E (2014) Representations of the Rook-Brauer algebra. Comm Algebra 42(1):423–443. 10.1080/00927872.2012.716120, URL https://doi.org/10.1080/00927872.2012.716120 Halverson and Ram [2005] Halverson T, Ram A (2005) Partition algebras. European J Combin 26(6):869–921. 10.1016/j.ejc.2004.06.005, URL http://dx.doi.org/10.1016/j.ejc.2004.06.005 Jacobson [1989] Jacobson N (1989) Basic algebra. II, 2nd edn. W. H. Freeman and Company, New York Jones [1994] Jones VFR (1994) The Potts model and the symmetric group. In: Subfactors (Kyuzeso, 1993). World Sci. Publ., River Edge, NJ, p 259–267 Kudryavtseva and Mazorchuk [2008] Kudryavtseva G, Mazorchuk V (2008) Schur-Weyl dualities for symmetric inverse semigroups. J Pure Appl Algebra 212(8):1987–1995. 10.1016/j.jpaa.2007.12.004, URL https://doi.org/10.1016/j.jpaa.2007.12.004 Lawson [1998] Lawson MV (1998) Inverse semigroups. World Scientific Publishing Co. Inc., River Edge, NJ, 10.1142/9789812816689, URL http://dx.doi.org/10.1142/9789812816689, the theory of partial symmetries Martin [1994] Martin P (1994) Temperley-Lieb algebras for nonplanar statistical mechanics—the partition algebra construction. J Knot Theory Ramifications 3(1):51–82. 10.1142/S0218216594000071, URL http://dx.doi.org/10.1142/S0218216594000071 Martin [1996] Martin P (1996) The structure of the partition algebras. J Algebra 183(2):319–358. 10.1006/jabr.1996.0223, URL http://dx.doi.org/10.1006/jabr.1996.0223 Martin and Mazorchuk [2014] Martin P, Mazorchuk V (2014) On the representation theory of partial Brauer algebras. Q J Math 65(1):225–247. 10.1093/qmath/has043, URL https://doi.org/10.1093/qmath/has043 Martin [1990] Martin PP (1990) Representations of graph Temperley-Lieb algebras. Publ Res Inst Math Sci 26(3):485–503. 10.2977/prims/1195170958, URL https://doi.org/10.2977/prims/1195170958 Martin [2008] Martin S (2008) Schur algebras and representation theory, Cambridge Tracts in Mathematics, vol 112. Cambridge University Press, Cambridge, reprint of the 1993 original Munn [1955] Munn WD (1955) On semigroup algebras. Proc Cambridge Philos Soc 51:1–15 Munn [1957a] Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Halverson T (2004) Representations of the q𝑞qitalic_q-rook monoid. J Algebra 273(1):227–251. 10.1016/j.jalgebra.2003.11.002, URL http://dx.doi.org/10.1016/j.jalgebra.2003.11.002 Halverson and delMas [2014] Halverson T, delMas E (2014) Representations of the Rook-Brauer algebra. Comm Algebra 42(1):423–443. 10.1080/00927872.2012.716120, URL https://doi.org/10.1080/00927872.2012.716120 Halverson and Ram [2005] Halverson T, Ram A (2005) Partition algebras. European J Combin 26(6):869–921. 10.1016/j.ejc.2004.06.005, URL http://dx.doi.org/10.1016/j.ejc.2004.06.005 Jacobson [1989] Jacobson N (1989) Basic algebra. II, 2nd edn. W. H. Freeman and Company, New York Jones [1994] Jones VFR (1994) The Potts model and the symmetric group. In: Subfactors (Kyuzeso, 1993). World Sci. Publ., River Edge, NJ, p 259–267 Kudryavtseva and Mazorchuk [2008] Kudryavtseva G, Mazorchuk V (2008) Schur-Weyl dualities for symmetric inverse semigroups. J Pure Appl Algebra 212(8):1987–1995. 10.1016/j.jpaa.2007.12.004, URL https://doi.org/10.1016/j.jpaa.2007.12.004 Lawson [1998] Lawson MV (1998) Inverse semigroups. World Scientific Publishing Co. Inc., River Edge, NJ, 10.1142/9789812816689, URL http://dx.doi.org/10.1142/9789812816689, the theory of partial symmetries Martin [1994] Martin P (1994) Temperley-Lieb algebras for nonplanar statistical mechanics—the partition algebra construction. J Knot Theory Ramifications 3(1):51–82. 10.1142/S0218216594000071, URL http://dx.doi.org/10.1142/S0218216594000071 Martin [1996] Martin P (1996) The structure of the partition algebras. J Algebra 183(2):319–358. 10.1006/jabr.1996.0223, URL http://dx.doi.org/10.1006/jabr.1996.0223 Martin and Mazorchuk [2014] Martin P, Mazorchuk V (2014) On the representation theory of partial Brauer algebras. Q J Math 65(1):225–247. 10.1093/qmath/has043, URL https://doi.org/10.1093/qmath/has043 Martin [1990] Martin PP (1990) Representations of graph Temperley-Lieb algebras. Publ Res Inst Math Sci 26(3):485–503. 10.2977/prims/1195170958, URL https://doi.org/10.2977/prims/1195170958 Martin [2008] Martin S (2008) Schur algebras and representation theory, Cambridge Tracts in Mathematics, vol 112. Cambridge University Press, Cambridge, reprint of the 1993 original Munn [1955] Munn WD (1955) On semigroup algebras. Proc Cambridge Philos Soc 51:1–15 Munn [1957a] Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Halverson T, delMas E (2014) Representations of the Rook-Brauer algebra. Comm Algebra 42(1):423–443. 10.1080/00927872.2012.716120, URL https://doi.org/10.1080/00927872.2012.716120 Halverson and Ram [2005] Halverson T, Ram A (2005) Partition algebras. European J Combin 26(6):869–921. 10.1016/j.ejc.2004.06.005, URL http://dx.doi.org/10.1016/j.ejc.2004.06.005 Jacobson [1989] Jacobson N (1989) Basic algebra. II, 2nd edn. W. H. Freeman and Company, New York Jones [1994] Jones VFR (1994) The Potts model and the symmetric group. In: Subfactors (Kyuzeso, 1993). World Sci. Publ., River Edge, NJ, p 259–267 Kudryavtseva and Mazorchuk [2008] Kudryavtseva G, Mazorchuk V (2008) Schur-Weyl dualities for symmetric inverse semigroups. J Pure Appl Algebra 212(8):1987–1995. 10.1016/j.jpaa.2007.12.004, URL https://doi.org/10.1016/j.jpaa.2007.12.004 Lawson [1998] Lawson MV (1998) Inverse semigroups. World Scientific Publishing Co. Inc., River Edge, NJ, 10.1142/9789812816689, URL http://dx.doi.org/10.1142/9789812816689, the theory of partial symmetries Martin [1994] Martin P (1994) Temperley-Lieb algebras for nonplanar statistical mechanics—the partition algebra construction. J Knot Theory Ramifications 3(1):51–82. 10.1142/S0218216594000071, URL http://dx.doi.org/10.1142/S0218216594000071 Martin [1996] Martin P (1996) The structure of the partition algebras. J Algebra 183(2):319–358. 10.1006/jabr.1996.0223, URL http://dx.doi.org/10.1006/jabr.1996.0223 Martin and Mazorchuk [2014] Martin P, Mazorchuk V (2014) On the representation theory of partial Brauer algebras. Q J Math 65(1):225–247. 10.1093/qmath/has043, URL https://doi.org/10.1093/qmath/has043 Martin [1990] Martin PP (1990) Representations of graph Temperley-Lieb algebras. Publ Res Inst Math Sci 26(3):485–503. 10.2977/prims/1195170958, URL https://doi.org/10.2977/prims/1195170958 Martin [2008] Martin S (2008) Schur algebras and representation theory, Cambridge Tracts in Mathematics, vol 112. Cambridge University Press, Cambridge, reprint of the 1993 original Munn [1955] Munn WD (1955) On semigroup algebras. Proc Cambridge Philos Soc 51:1–15 Munn [1957a] Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Halverson T, Ram A (2005) Partition algebras. European J Combin 26(6):869–921. 10.1016/j.ejc.2004.06.005, URL http://dx.doi.org/10.1016/j.ejc.2004.06.005 Jacobson [1989] Jacobson N (1989) Basic algebra. II, 2nd edn. W. H. Freeman and Company, New York Jones [1994] Jones VFR (1994) The Potts model and the symmetric group. In: Subfactors (Kyuzeso, 1993). World Sci. Publ., River Edge, NJ, p 259–267 Kudryavtseva and Mazorchuk [2008] Kudryavtseva G, Mazorchuk V (2008) Schur-Weyl dualities for symmetric inverse semigroups. J Pure Appl Algebra 212(8):1987–1995. 10.1016/j.jpaa.2007.12.004, URL https://doi.org/10.1016/j.jpaa.2007.12.004 Lawson [1998] Lawson MV (1998) Inverse semigroups. World Scientific Publishing Co. Inc., River Edge, NJ, 10.1142/9789812816689, URL http://dx.doi.org/10.1142/9789812816689, the theory of partial symmetries Martin [1994] Martin P (1994) Temperley-Lieb algebras for nonplanar statistical mechanics—the partition algebra construction. J Knot Theory Ramifications 3(1):51–82. 10.1142/S0218216594000071, URL http://dx.doi.org/10.1142/S0218216594000071 Martin [1996] Martin P (1996) The structure of the partition algebras. J Algebra 183(2):319–358. 10.1006/jabr.1996.0223, URL http://dx.doi.org/10.1006/jabr.1996.0223 Martin and Mazorchuk [2014] Martin P, Mazorchuk V (2014) On the representation theory of partial Brauer algebras. Q J Math 65(1):225–247. 10.1093/qmath/has043, URL https://doi.org/10.1093/qmath/has043 Martin [1990] Martin PP (1990) Representations of graph Temperley-Lieb algebras. Publ Res Inst Math Sci 26(3):485–503. 10.2977/prims/1195170958, URL https://doi.org/10.2977/prims/1195170958 Martin [2008] Martin S (2008) Schur algebras and representation theory, Cambridge Tracts in Mathematics, vol 112. Cambridge University Press, Cambridge, reprint of the 1993 original Munn [1955] Munn WD (1955) On semigroup algebras. Proc Cambridge Philos Soc 51:1–15 Munn [1957a] Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Jacobson N (1989) Basic algebra. II, 2nd edn. W. H. Freeman and Company, New York Jones [1994] Jones VFR (1994) The Potts model and the symmetric group. In: Subfactors (Kyuzeso, 1993). World Sci. Publ., River Edge, NJ, p 259–267 Kudryavtseva and Mazorchuk [2008] Kudryavtseva G, Mazorchuk V (2008) Schur-Weyl dualities for symmetric inverse semigroups. J Pure Appl Algebra 212(8):1987–1995. 10.1016/j.jpaa.2007.12.004, URL https://doi.org/10.1016/j.jpaa.2007.12.004 Lawson [1998] Lawson MV (1998) Inverse semigroups. World Scientific Publishing Co. Inc., River Edge, NJ, 10.1142/9789812816689, URL http://dx.doi.org/10.1142/9789812816689, the theory of partial symmetries Martin [1994] Martin P (1994) Temperley-Lieb algebras for nonplanar statistical mechanics—the partition algebra construction. J Knot Theory Ramifications 3(1):51–82. 10.1142/S0218216594000071, URL http://dx.doi.org/10.1142/S0218216594000071 Martin [1996] Martin P (1996) The structure of the partition algebras. J Algebra 183(2):319–358. 10.1006/jabr.1996.0223, URL http://dx.doi.org/10.1006/jabr.1996.0223 Martin and Mazorchuk [2014] Martin P, Mazorchuk V (2014) On the representation theory of partial Brauer algebras. Q J Math 65(1):225–247. 10.1093/qmath/has043, URL https://doi.org/10.1093/qmath/has043 Martin [1990] Martin PP (1990) Representations of graph Temperley-Lieb algebras. Publ Res Inst Math Sci 26(3):485–503. 10.2977/prims/1195170958, URL https://doi.org/10.2977/prims/1195170958 Martin [2008] Martin S (2008) Schur algebras and representation theory, Cambridge Tracts in Mathematics, vol 112. Cambridge University Press, Cambridge, reprint of the 1993 original Munn [1955] Munn WD (1955) On semigroup algebras. Proc Cambridge Philos Soc 51:1–15 Munn [1957a] Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Jones VFR (1994) The Potts model and the symmetric group. In: Subfactors (Kyuzeso, 1993). World Sci. Publ., River Edge, NJ, p 259–267 Kudryavtseva and Mazorchuk [2008] Kudryavtseva G, Mazorchuk V (2008) Schur-Weyl dualities for symmetric inverse semigroups. J Pure Appl Algebra 212(8):1987–1995. 10.1016/j.jpaa.2007.12.004, URL https://doi.org/10.1016/j.jpaa.2007.12.004 Lawson [1998] Lawson MV (1998) Inverse semigroups. World Scientific Publishing Co. Inc., River Edge, NJ, 10.1142/9789812816689, URL http://dx.doi.org/10.1142/9789812816689, the theory of partial symmetries Martin [1994] Martin P (1994) Temperley-Lieb algebras for nonplanar statistical mechanics—the partition algebra construction. J Knot Theory Ramifications 3(1):51–82. 10.1142/S0218216594000071, URL http://dx.doi.org/10.1142/S0218216594000071 Martin [1996] Martin P (1996) The structure of the partition algebras. J Algebra 183(2):319–358. 10.1006/jabr.1996.0223, URL http://dx.doi.org/10.1006/jabr.1996.0223 Martin and Mazorchuk [2014] Martin P, Mazorchuk V (2014) On the representation theory of partial Brauer algebras. Q J Math 65(1):225–247. 10.1093/qmath/has043, URL https://doi.org/10.1093/qmath/has043 Martin [1990] Martin PP (1990) Representations of graph Temperley-Lieb algebras. Publ Res Inst Math Sci 26(3):485–503. 10.2977/prims/1195170958, URL https://doi.org/10.2977/prims/1195170958 Martin [2008] Martin S (2008) Schur algebras and representation theory, Cambridge Tracts in Mathematics, vol 112. Cambridge University Press, Cambridge, reprint of the 1993 original Munn [1955] Munn WD (1955) On semigroup algebras. Proc Cambridge Philos Soc 51:1–15 Munn [1957a] Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Kudryavtseva G, Mazorchuk V (2008) Schur-Weyl dualities for symmetric inverse semigroups. J Pure Appl Algebra 212(8):1987–1995. 10.1016/j.jpaa.2007.12.004, URL https://doi.org/10.1016/j.jpaa.2007.12.004 Lawson [1998] Lawson MV (1998) Inverse semigroups. World Scientific Publishing Co. Inc., River Edge, NJ, 10.1142/9789812816689, URL http://dx.doi.org/10.1142/9789812816689, the theory of partial symmetries Martin [1994] Martin P (1994) Temperley-Lieb algebras for nonplanar statistical mechanics—the partition algebra construction. J Knot Theory Ramifications 3(1):51–82. 10.1142/S0218216594000071, URL http://dx.doi.org/10.1142/S0218216594000071 Martin [1996] Martin P (1996) The structure of the partition algebras. J Algebra 183(2):319–358. 10.1006/jabr.1996.0223, URL http://dx.doi.org/10.1006/jabr.1996.0223 Martin and Mazorchuk [2014] Martin P, Mazorchuk V (2014) On the representation theory of partial Brauer algebras. Q J Math 65(1):225–247. 10.1093/qmath/has043, URL https://doi.org/10.1093/qmath/has043 Martin [1990] Martin PP (1990) Representations of graph Temperley-Lieb algebras. Publ Res Inst Math Sci 26(3):485–503. 10.2977/prims/1195170958, URL https://doi.org/10.2977/prims/1195170958 Martin [2008] Martin S (2008) Schur algebras and representation theory, Cambridge Tracts in Mathematics, vol 112. Cambridge University Press, Cambridge, reprint of the 1993 original Munn [1955] Munn WD (1955) On semigroup algebras. Proc Cambridge Philos Soc 51:1–15 Munn [1957a] Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Lawson MV (1998) Inverse semigroups. World Scientific Publishing Co. Inc., River Edge, NJ, 10.1142/9789812816689, URL http://dx.doi.org/10.1142/9789812816689, the theory of partial symmetries Martin [1994] Martin P (1994) Temperley-Lieb algebras for nonplanar statistical mechanics—the partition algebra construction. J Knot Theory Ramifications 3(1):51–82. 10.1142/S0218216594000071, URL http://dx.doi.org/10.1142/S0218216594000071 Martin [1996] Martin P (1996) The structure of the partition algebras. J Algebra 183(2):319–358. 10.1006/jabr.1996.0223, URL http://dx.doi.org/10.1006/jabr.1996.0223 Martin and Mazorchuk [2014] Martin P, Mazorchuk V (2014) On the representation theory of partial Brauer algebras. Q J Math 65(1):225–247. 10.1093/qmath/has043, URL https://doi.org/10.1093/qmath/has043 Martin [1990] Martin PP (1990) Representations of graph Temperley-Lieb algebras. Publ Res Inst Math Sci 26(3):485–503. 10.2977/prims/1195170958, URL https://doi.org/10.2977/prims/1195170958 Martin [2008] Martin S (2008) Schur algebras and representation theory, Cambridge Tracts in Mathematics, vol 112. Cambridge University Press, Cambridge, reprint of the 1993 original Munn [1955] Munn WD (1955) On semigroup algebras. Proc Cambridge Philos Soc 51:1–15 Munn [1957a] Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Martin P (1994) Temperley-Lieb algebras for nonplanar statistical mechanics—the partition algebra construction. J Knot Theory Ramifications 3(1):51–82. 10.1142/S0218216594000071, URL http://dx.doi.org/10.1142/S0218216594000071 Martin [1996] Martin P (1996) The structure of the partition algebras. J Algebra 183(2):319–358. 10.1006/jabr.1996.0223, URL http://dx.doi.org/10.1006/jabr.1996.0223 Martin and Mazorchuk [2014] Martin P, Mazorchuk V (2014) On the representation theory of partial Brauer algebras. Q J Math 65(1):225–247. 10.1093/qmath/has043, URL https://doi.org/10.1093/qmath/has043 Martin [1990] Martin PP (1990) Representations of graph Temperley-Lieb algebras. Publ Res Inst Math Sci 26(3):485–503. 10.2977/prims/1195170958, URL https://doi.org/10.2977/prims/1195170958 Martin [2008] Martin S (2008) Schur algebras and representation theory, Cambridge Tracts in Mathematics, vol 112. Cambridge University Press, Cambridge, reprint of the 1993 original Munn [1955] Munn WD (1955) On semigroup algebras. Proc Cambridge Philos Soc 51:1–15 Munn [1957a] Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Martin P (1996) The structure of the partition algebras. J Algebra 183(2):319–358. 10.1006/jabr.1996.0223, URL http://dx.doi.org/10.1006/jabr.1996.0223 Martin and Mazorchuk [2014] Martin P, Mazorchuk V (2014) On the representation theory of partial Brauer algebras. Q J Math 65(1):225–247. 10.1093/qmath/has043, URL https://doi.org/10.1093/qmath/has043 Martin [1990] Martin PP (1990) Representations of graph Temperley-Lieb algebras. Publ Res Inst Math Sci 26(3):485–503. 10.2977/prims/1195170958, URL https://doi.org/10.2977/prims/1195170958 Martin [2008] Martin S (2008) Schur algebras and representation theory, Cambridge Tracts in Mathematics, vol 112. Cambridge University Press, Cambridge, reprint of the 1993 original Munn [1955] Munn WD (1955) On semigroup algebras. Proc Cambridge Philos Soc 51:1–15 Munn [1957a] Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Martin P, Mazorchuk V (2014) On the representation theory of partial Brauer algebras. Q J Math 65(1):225–247. 10.1093/qmath/has043, URL https://doi.org/10.1093/qmath/has043 Martin [1990] Martin PP (1990) Representations of graph Temperley-Lieb algebras. Publ Res Inst Math Sci 26(3):485–503. 10.2977/prims/1195170958, URL https://doi.org/10.2977/prims/1195170958 Martin [2008] Martin S (2008) Schur algebras and representation theory, Cambridge Tracts in Mathematics, vol 112. Cambridge University Press, Cambridge, reprint of the 1993 original Munn [1955] Munn WD (1955) On semigroup algebras. Proc Cambridge Philos Soc 51:1–15 Munn [1957a] Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Martin PP (1990) Representations of graph Temperley-Lieb algebras. Publ Res Inst Math Sci 26(3):485–503. 10.2977/prims/1195170958, URL https://doi.org/10.2977/prims/1195170958 Martin [2008] Martin S (2008) Schur algebras and representation theory, Cambridge Tracts in Mathematics, vol 112. Cambridge University Press, Cambridge, reprint of the 1993 original Munn [1955] Munn WD (1955) On semigroup algebras. Proc Cambridge Philos Soc 51:1–15 Munn [1957a] Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Martin S (2008) Schur algebras and representation theory, Cambridge Tracts in Mathematics, vol 112. Cambridge University Press, Cambridge, reprint of the 1993 original Munn [1955] Munn WD (1955) On semigroup algebras. Proc Cambridge Philos Soc 51:1–15 Munn [1957a] Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Munn WD (1955) On semigroup algebras. Proc Cambridge Philos Soc 51:1–15 Munn [1957a] Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J.
- Green JA (1980) Polynomial representations of GLnsubscriptGL𝑛{\rm GL}_{n}roman_GL start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT, Lecture Notes in Mathematics, vol 830. Springer-Verlag, Berlin Grood [2006] Grood C (2006) The rook partition algebra. J Combin Theory Ser A 113(2):325–351. 10.1016/j.jcta.2005.03.006, URL https://doi.org/10.1016/j.jcta.2005.03.006 Halverson [1996] Halverson T (1996) Characters of the centralizer algebras of mixed tensor representations of Gl(r,ℂ)Gl𝑟ℂ{\rm Gl}(r,\mathbb{C})roman_Gl ( italic_r , blackboard_C ) and the quantum group 𝔘q(ℊl(r,ℂ))subscript𝔘𝑞ℊ𝑙𝑟ℂ\mathfrak{U}_{q}({\mathcal{g}l}(r,\mathbb{C}))fraktur_U start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ( caligraphic_g italic_l ( italic_r , blackboard_C ) ). Pacific J Math 174(2):359–410. URL http://projecteuclid.org/euclid.pjm/1102365176 Halverson [2004] Halverson T (2004) Representations of the q𝑞qitalic_q-rook monoid. J Algebra 273(1):227–251. 10.1016/j.jalgebra.2003.11.002, URL http://dx.doi.org/10.1016/j.jalgebra.2003.11.002 Halverson and delMas [2014] Halverson T, delMas E (2014) Representations of the Rook-Brauer algebra. Comm Algebra 42(1):423–443. 10.1080/00927872.2012.716120, URL https://doi.org/10.1080/00927872.2012.716120 Halverson and Ram [2005] Halverson T, Ram A (2005) Partition algebras. European J Combin 26(6):869–921. 10.1016/j.ejc.2004.06.005, URL http://dx.doi.org/10.1016/j.ejc.2004.06.005 Jacobson [1989] Jacobson N (1989) Basic algebra. II, 2nd edn. W. H. Freeman and Company, New York Jones [1994] Jones VFR (1994) The Potts model and the symmetric group. In: Subfactors (Kyuzeso, 1993). World Sci. Publ., River Edge, NJ, p 259–267 Kudryavtseva and Mazorchuk [2008] Kudryavtseva G, Mazorchuk V (2008) Schur-Weyl dualities for symmetric inverse semigroups. J Pure Appl Algebra 212(8):1987–1995. 10.1016/j.jpaa.2007.12.004, URL https://doi.org/10.1016/j.jpaa.2007.12.004 Lawson [1998] Lawson MV (1998) Inverse semigroups. World Scientific Publishing Co. Inc., River Edge, NJ, 10.1142/9789812816689, URL http://dx.doi.org/10.1142/9789812816689, the theory of partial symmetries Martin [1994] Martin P (1994) Temperley-Lieb algebras for nonplanar statistical mechanics—the partition algebra construction. J Knot Theory Ramifications 3(1):51–82. 10.1142/S0218216594000071, URL http://dx.doi.org/10.1142/S0218216594000071 Martin [1996] Martin P (1996) The structure of the partition algebras. J Algebra 183(2):319–358. 10.1006/jabr.1996.0223, URL http://dx.doi.org/10.1006/jabr.1996.0223 Martin and Mazorchuk [2014] Martin P, Mazorchuk V (2014) On the representation theory of partial Brauer algebras. Q J Math 65(1):225–247. 10.1093/qmath/has043, URL https://doi.org/10.1093/qmath/has043 Martin [1990] Martin PP (1990) Representations of graph Temperley-Lieb algebras. Publ Res Inst Math Sci 26(3):485–503. 10.2977/prims/1195170958, URL https://doi.org/10.2977/prims/1195170958 Martin [2008] Martin S (2008) Schur algebras and representation theory, Cambridge Tracts in Mathematics, vol 112. Cambridge University Press, Cambridge, reprint of the 1993 original Munn [1955] Munn WD (1955) On semigroup algebras. Proc Cambridge Philos Soc 51:1–15 Munn [1957a] Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Grood C (2006) The rook partition algebra. J Combin Theory Ser A 113(2):325–351. 10.1016/j.jcta.2005.03.006, URL https://doi.org/10.1016/j.jcta.2005.03.006 Halverson [1996] Halverson T (1996) Characters of the centralizer algebras of mixed tensor representations of Gl(r,ℂ)Gl𝑟ℂ{\rm Gl}(r,\mathbb{C})roman_Gl ( italic_r , blackboard_C ) and the quantum group 𝔘q(ℊl(r,ℂ))subscript𝔘𝑞ℊ𝑙𝑟ℂ\mathfrak{U}_{q}({\mathcal{g}l}(r,\mathbb{C}))fraktur_U start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ( caligraphic_g italic_l ( italic_r , blackboard_C ) ). Pacific J Math 174(2):359–410. URL http://projecteuclid.org/euclid.pjm/1102365176 Halverson [2004] Halverson T (2004) Representations of the q𝑞qitalic_q-rook monoid. J Algebra 273(1):227–251. 10.1016/j.jalgebra.2003.11.002, URL http://dx.doi.org/10.1016/j.jalgebra.2003.11.002 Halverson and delMas [2014] Halverson T, delMas E (2014) Representations of the Rook-Brauer algebra. Comm Algebra 42(1):423–443. 10.1080/00927872.2012.716120, URL https://doi.org/10.1080/00927872.2012.716120 Halverson and Ram [2005] Halverson T, Ram A (2005) Partition algebras. European J Combin 26(6):869–921. 10.1016/j.ejc.2004.06.005, URL http://dx.doi.org/10.1016/j.ejc.2004.06.005 Jacobson [1989] Jacobson N (1989) Basic algebra. II, 2nd edn. W. H. Freeman and Company, New York Jones [1994] Jones VFR (1994) The Potts model and the symmetric group. In: Subfactors (Kyuzeso, 1993). World Sci. Publ., River Edge, NJ, p 259–267 Kudryavtseva and Mazorchuk [2008] Kudryavtseva G, Mazorchuk V (2008) Schur-Weyl dualities for symmetric inverse semigroups. J Pure Appl Algebra 212(8):1987–1995. 10.1016/j.jpaa.2007.12.004, URL https://doi.org/10.1016/j.jpaa.2007.12.004 Lawson [1998] Lawson MV (1998) Inverse semigroups. World Scientific Publishing Co. Inc., River Edge, NJ, 10.1142/9789812816689, URL http://dx.doi.org/10.1142/9789812816689, the theory of partial symmetries Martin [1994] Martin P (1994) Temperley-Lieb algebras for nonplanar statistical mechanics—the partition algebra construction. J Knot Theory Ramifications 3(1):51–82. 10.1142/S0218216594000071, URL http://dx.doi.org/10.1142/S0218216594000071 Martin [1996] Martin P (1996) The structure of the partition algebras. J Algebra 183(2):319–358. 10.1006/jabr.1996.0223, URL http://dx.doi.org/10.1006/jabr.1996.0223 Martin and Mazorchuk [2014] Martin P, Mazorchuk V (2014) On the representation theory of partial Brauer algebras. Q J Math 65(1):225–247. 10.1093/qmath/has043, URL https://doi.org/10.1093/qmath/has043 Martin [1990] Martin PP (1990) Representations of graph Temperley-Lieb algebras. Publ Res Inst Math Sci 26(3):485–503. 10.2977/prims/1195170958, URL https://doi.org/10.2977/prims/1195170958 Martin [2008] Martin S (2008) Schur algebras and representation theory, Cambridge Tracts in Mathematics, vol 112. Cambridge University Press, Cambridge, reprint of the 1993 original Munn [1955] Munn WD (1955) On semigroup algebras. Proc Cambridge Philos Soc 51:1–15 Munn [1957a] Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Halverson T (1996) Characters of the centralizer algebras of mixed tensor representations of Gl(r,ℂ)Gl𝑟ℂ{\rm Gl}(r,\mathbb{C})roman_Gl ( italic_r , blackboard_C ) and the quantum group 𝔘q(ℊl(r,ℂ))subscript𝔘𝑞ℊ𝑙𝑟ℂ\mathfrak{U}_{q}({\mathcal{g}l}(r,\mathbb{C}))fraktur_U start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ( caligraphic_g italic_l ( italic_r , blackboard_C ) ). Pacific J Math 174(2):359–410. URL http://projecteuclid.org/euclid.pjm/1102365176 Halverson [2004] Halverson T (2004) Representations of the q𝑞qitalic_q-rook monoid. J Algebra 273(1):227–251. 10.1016/j.jalgebra.2003.11.002, URL http://dx.doi.org/10.1016/j.jalgebra.2003.11.002 Halverson and delMas [2014] Halverson T, delMas E (2014) Representations of the Rook-Brauer algebra. Comm Algebra 42(1):423–443. 10.1080/00927872.2012.716120, URL https://doi.org/10.1080/00927872.2012.716120 Halverson and Ram [2005] Halverson T, Ram A (2005) Partition algebras. European J Combin 26(6):869–921. 10.1016/j.ejc.2004.06.005, URL http://dx.doi.org/10.1016/j.ejc.2004.06.005 Jacobson [1989] Jacobson N (1989) Basic algebra. II, 2nd edn. W. H. Freeman and Company, New York Jones [1994] Jones VFR (1994) The Potts model and the symmetric group. In: Subfactors (Kyuzeso, 1993). World Sci. Publ., River Edge, NJ, p 259–267 Kudryavtseva and Mazorchuk [2008] Kudryavtseva G, Mazorchuk V (2008) Schur-Weyl dualities for symmetric inverse semigroups. J Pure Appl Algebra 212(8):1987–1995. 10.1016/j.jpaa.2007.12.004, URL https://doi.org/10.1016/j.jpaa.2007.12.004 Lawson [1998] Lawson MV (1998) Inverse semigroups. World Scientific Publishing Co. Inc., River Edge, NJ, 10.1142/9789812816689, URL http://dx.doi.org/10.1142/9789812816689, the theory of partial symmetries Martin [1994] Martin P (1994) Temperley-Lieb algebras for nonplanar statistical mechanics—the partition algebra construction. J Knot Theory Ramifications 3(1):51–82. 10.1142/S0218216594000071, URL http://dx.doi.org/10.1142/S0218216594000071 Martin [1996] Martin P (1996) The structure of the partition algebras. J Algebra 183(2):319–358. 10.1006/jabr.1996.0223, URL http://dx.doi.org/10.1006/jabr.1996.0223 Martin and Mazorchuk [2014] Martin P, Mazorchuk V (2014) On the representation theory of partial Brauer algebras. Q J Math 65(1):225–247. 10.1093/qmath/has043, URL https://doi.org/10.1093/qmath/has043 Martin [1990] Martin PP (1990) Representations of graph Temperley-Lieb algebras. Publ Res Inst Math Sci 26(3):485–503. 10.2977/prims/1195170958, URL https://doi.org/10.2977/prims/1195170958 Martin [2008] Martin S (2008) Schur algebras and representation theory, Cambridge Tracts in Mathematics, vol 112. Cambridge University Press, Cambridge, reprint of the 1993 original Munn [1955] Munn WD (1955) On semigroup algebras. Proc Cambridge Philos Soc 51:1–15 Munn [1957a] Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Halverson T (2004) Representations of the q𝑞qitalic_q-rook monoid. J Algebra 273(1):227–251. 10.1016/j.jalgebra.2003.11.002, URL http://dx.doi.org/10.1016/j.jalgebra.2003.11.002 Halverson and delMas [2014] Halverson T, delMas E (2014) Representations of the Rook-Brauer algebra. Comm Algebra 42(1):423–443. 10.1080/00927872.2012.716120, URL https://doi.org/10.1080/00927872.2012.716120 Halverson and Ram [2005] Halverson T, Ram A (2005) Partition algebras. European J Combin 26(6):869–921. 10.1016/j.ejc.2004.06.005, URL http://dx.doi.org/10.1016/j.ejc.2004.06.005 Jacobson [1989] Jacobson N (1989) Basic algebra. II, 2nd edn. W. H. Freeman and Company, New York Jones [1994] Jones VFR (1994) The Potts model and the symmetric group. In: Subfactors (Kyuzeso, 1993). World Sci. Publ., River Edge, NJ, p 259–267 Kudryavtseva and Mazorchuk [2008] Kudryavtseva G, Mazorchuk V (2008) Schur-Weyl dualities for symmetric inverse semigroups. J Pure Appl Algebra 212(8):1987–1995. 10.1016/j.jpaa.2007.12.004, URL https://doi.org/10.1016/j.jpaa.2007.12.004 Lawson [1998] Lawson MV (1998) Inverse semigroups. World Scientific Publishing Co. Inc., River Edge, NJ, 10.1142/9789812816689, URL http://dx.doi.org/10.1142/9789812816689, the theory of partial symmetries Martin [1994] Martin P (1994) Temperley-Lieb algebras for nonplanar statistical mechanics—the partition algebra construction. J Knot Theory Ramifications 3(1):51–82. 10.1142/S0218216594000071, URL http://dx.doi.org/10.1142/S0218216594000071 Martin [1996] Martin P (1996) The structure of the partition algebras. J Algebra 183(2):319–358. 10.1006/jabr.1996.0223, URL http://dx.doi.org/10.1006/jabr.1996.0223 Martin and Mazorchuk [2014] Martin P, Mazorchuk V (2014) On the representation theory of partial Brauer algebras. Q J Math 65(1):225–247. 10.1093/qmath/has043, URL https://doi.org/10.1093/qmath/has043 Martin [1990] Martin PP (1990) Representations of graph Temperley-Lieb algebras. Publ Res Inst Math Sci 26(3):485–503. 10.2977/prims/1195170958, URL https://doi.org/10.2977/prims/1195170958 Martin [2008] Martin S (2008) Schur algebras and representation theory, Cambridge Tracts in Mathematics, vol 112. Cambridge University Press, Cambridge, reprint of the 1993 original Munn [1955] Munn WD (1955) On semigroup algebras. Proc Cambridge Philos Soc 51:1–15 Munn [1957a] Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Halverson T, delMas E (2014) Representations of the Rook-Brauer algebra. Comm Algebra 42(1):423–443. 10.1080/00927872.2012.716120, URL https://doi.org/10.1080/00927872.2012.716120 Halverson and Ram [2005] Halverson T, Ram A (2005) Partition algebras. European J Combin 26(6):869–921. 10.1016/j.ejc.2004.06.005, URL http://dx.doi.org/10.1016/j.ejc.2004.06.005 Jacobson [1989] Jacobson N (1989) Basic algebra. II, 2nd edn. W. H. Freeman and Company, New York Jones [1994] Jones VFR (1994) The Potts model and the symmetric group. In: Subfactors (Kyuzeso, 1993). World Sci. Publ., River Edge, NJ, p 259–267 Kudryavtseva and Mazorchuk [2008] Kudryavtseva G, Mazorchuk V (2008) Schur-Weyl dualities for symmetric inverse semigroups. J Pure Appl Algebra 212(8):1987–1995. 10.1016/j.jpaa.2007.12.004, URL https://doi.org/10.1016/j.jpaa.2007.12.004 Lawson [1998] Lawson MV (1998) Inverse semigroups. World Scientific Publishing Co. Inc., River Edge, NJ, 10.1142/9789812816689, URL http://dx.doi.org/10.1142/9789812816689, the theory of partial symmetries Martin [1994] Martin P (1994) Temperley-Lieb algebras for nonplanar statistical mechanics—the partition algebra construction. J Knot Theory Ramifications 3(1):51–82. 10.1142/S0218216594000071, URL http://dx.doi.org/10.1142/S0218216594000071 Martin [1996] Martin P (1996) The structure of the partition algebras. J Algebra 183(2):319–358. 10.1006/jabr.1996.0223, URL http://dx.doi.org/10.1006/jabr.1996.0223 Martin and Mazorchuk [2014] Martin P, Mazorchuk V (2014) On the representation theory of partial Brauer algebras. Q J Math 65(1):225–247. 10.1093/qmath/has043, URL https://doi.org/10.1093/qmath/has043 Martin [1990] Martin PP (1990) Representations of graph Temperley-Lieb algebras. Publ Res Inst Math Sci 26(3):485–503. 10.2977/prims/1195170958, URL https://doi.org/10.2977/prims/1195170958 Martin [2008] Martin S (2008) Schur algebras and representation theory, Cambridge Tracts in Mathematics, vol 112. Cambridge University Press, Cambridge, reprint of the 1993 original Munn [1955] Munn WD (1955) On semigroup algebras. Proc Cambridge Philos Soc 51:1–15 Munn [1957a] Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Halverson T, Ram A (2005) Partition algebras. European J Combin 26(6):869–921. 10.1016/j.ejc.2004.06.005, URL http://dx.doi.org/10.1016/j.ejc.2004.06.005 Jacobson [1989] Jacobson N (1989) Basic algebra. II, 2nd edn. W. H. Freeman and Company, New York Jones [1994] Jones VFR (1994) The Potts model and the symmetric group. In: Subfactors (Kyuzeso, 1993). World Sci. Publ., River Edge, NJ, p 259–267 Kudryavtseva and Mazorchuk [2008] Kudryavtseva G, Mazorchuk V (2008) Schur-Weyl dualities for symmetric inverse semigroups. J Pure Appl Algebra 212(8):1987–1995. 10.1016/j.jpaa.2007.12.004, URL https://doi.org/10.1016/j.jpaa.2007.12.004 Lawson [1998] Lawson MV (1998) Inverse semigroups. World Scientific Publishing Co. Inc., River Edge, NJ, 10.1142/9789812816689, URL http://dx.doi.org/10.1142/9789812816689, the theory of partial symmetries Martin [1994] Martin P (1994) Temperley-Lieb algebras for nonplanar statistical mechanics—the partition algebra construction. J Knot Theory Ramifications 3(1):51–82. 10.1142/S0218216594000071, URL http://dx.doi.org/10.1142/S0218216594000071 Martin [1996] Martin P (1996) The structure of the partition algebras. J Algebra 183(2):319–358. 10.1006/jabr.1996.0223, URL http://dx.doi.org/10.1006/jabr.1996.0223 Martin and Mazorchuk [2014] Martin P, Mazorchuk V (2014) On the representation theory of partial Brauer algebras. Q J Math 65(1):225–247. 10.1093/qmath/has043, URL https://doi.org/10.1093/qmath/has043 Martin [1990] Martin PP (1990) Representations of graph Temperley-Lieb algebras. Publ Res Inst Math Sci 26(3):485–503. 10.2977/prims/1195170958, URL https://doi.org/10.2977/prims/1195170958 Martin [2008] Martin S (2008) Schur algebras and representation theory, Cambridge Tracts in Mathematics, vol 112. Cambridge University Press, Cambridge, reprint of the 1993 original Munn [1955] Munn WD (1955) On semigroup algebras. Proc Cambridge Philos Soc 51:1–15 Munn [1957a] Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Jacobson N (1989) Basic algebra. II, 2nd edn. W. H. Freeman and Company, New York Jones [1994] Jones VFR (1994) The Potts model and the symmetric group. In: Subfactors (Kyuzeso, 1993). World Sci. Publ., River Edge, NJ, p 259–267 Kudryavtseva and Mazorchuk [2008] Kudryavtseva G, Mazorchuk V (2008) Schur-Weyl dualities for symmetric inverse semigroups. J Pure Appl Algebra 212(8):1987–1995. 10.1016/j.jpaa.2007.12.004, URL https://doi.org/10.1016/j.jpaa.2007.12.004 Lawson [1998] Lawson MV (1998) Inverse semigroups. World Scientific Publishing Co. Inc., River Edge, NJ, 10.1142/9789812816689, URL http://dx.doi.org/10.1142/9789812816689, the theory of partial symmetries Martin [1994] Martin P (1994) Temperley-Lieb algebras for nonplanar statistical mechanics—the partition algebra construction. J Knot Theory Ramifications 3(1):51–82. 10.1142/S0218216594000071, URL http://dx.doi.org/10.1142/S0218216594000071 Martin [1996] Martin P (1996) The structure of the partition algebras. J Algebra 183(2):319–358. 10.1006/jabr.1996.0223, URL http://dx.doi.org/10.1006/jabr.1996.0223 Martin and Mazorchuk [2014] Martin P, Mazorchuk V (2014) On the representation theory of partial Brauer algebras. Q J Math 65(1):225–247. 10.1093/qmath/has043, URL https://doi.org/10.1093/qmath/has043 Martin [1990] Martin PP (1990) Representations of graph Temperley-Lieb algebras. Publ Res Inst Math Sci 26(3):485–503. 10.2977/prims/1195170958, URL https://doi.org/10.2977/prims/1195170958 Martin [2008] Martin S (2008) Schur algebras and representation theory, Cambridge Tracts in Mathematics, vol 112. Cambridge University Press, Cambridge, reprint of the 1993 original Munn [1955] Munn WD (1955) On semigroup algebras. Proc Cambridge Philos Soc 51:1–15 Munn [1957a] Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Jones VFR (1994) The Potts model and the symmetric group. In: Subfactors (Kyuzeso, 1993). World Sci. Publ., River Edge, NJ, p 259–267 Kudryavtseva and Mazorchuk [2008] Kudryavtseva G, Mazorchuk V (2008) Schur-Weyl dualities for symmetric inverse semigroups. J Pure Appl Algebra 212(8):1987–1995. 10.1016/j.jpaa.2007.12.004, URL https://doi.org/10.1016/j.jpaa.2007.12.004 Lawson [1998] Lawson MV (1998) Inverse semigroups. World Scientific Publishing Co. Inc., River Edge, NJ, 10.1142/9789812816689, URL http://dx.doi.org/10.1142/9789812816689, the theory of partial symmetries Martin [1994] Martin P (1994) Temperley-Lieb algebras for nonplanar statistical mechanics—the partition algebra construction. J Knot Theory Ramifications 3(1):51–82. 10.1142/S0218216594000071, URL http://dx.doi.org/10.1142/S0218216594000071 Martin [1996] Martin P (1996) The structure of the partition algebras. J Algebra 183(2):319–358. 10.1006/jabr.1996.0223, URL http://dx.doi.org/10.1006/jabr.1996.0223 Martin and Mazorchuk [2014] Martin P, Mazorchuk V (2014) On the representation theory of partial Brauer algebras. Q J Math 65(1):225–247. 10.1093/qmath/has043, URL https://doi.org/10.1093/qmath/has043 Martin [1990] Martin PP (1990) Representations of graph Temperley-Lieb algebras. Publ Res Inst Math Sci 26(3):485–503. 10.2977/prims/1195170958, URL https://doi.org/10.2977/prims/1195170958 Martin [2008] Martin S (2008) Schur algebras and representation theory, Cambridge Tracts in Mathematics, vol 112. Cambridge University Press, Cambridge, reprint of the 1993 original Munn [1955] Munn WD (1955) On semigroup algebras. Proc Cambridge Philos Soc 51:1–15 Munn [1957a] Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Kudryavtseva G, Mazorchuk V (2008) Schur-Weyl dualities for symmetric inverse semigroups. J Pure Appl Algebra 212(8):1987–1995. 10.1016/j.jpaa.2007.12.004, URL https://doi.org/10.1016/j.jpaa.2007.12.004 Lawson [1998] Lawson MV (1998) Inverse semigroups. World Scientific Publishing Co. Inc., River Edge, NJ, 10.1142/9789812816689, URL http://dx.doi.org/10.1142/9789812816689, the theory of partial symmetries Martin [1994] Martin P (1994) Temperley-Lieb algebras for nonplanar statistical mechanics—the partition algebra construction. J Knot Theory Ramifications 3(1):51–82. 10.1142/S0218216594000071, URL http://dx.doi.org/10.1142/S0218216594000071 Martin [1996] Martin P (1996) The structure of the partition algebras. J Algebra 183(2):319–358. 10.1006/jabr.1996.0223, URL http://dx.doi.org/10.1006/jabr.1996.0223 Martin and Mazorchuk [2014] Martin P, Mazorchuk V (2014) On the representation theory of partial Brauer algebras. Q J Math 65(1):225–247. 10.1093/qmath/has043, URL https://doi.org/10.1093/qmath/has043 Martin [1990] Martin PP (1990) Representations of graph Temperley-Lieb algebras. Publ Res Inst Math Sci 26(3):485–503. 10.2977/prims/1195170958, URL https://doi.org/10.2977/prims/1195170958 Martin [2008] Martin S (2008) Schur algebras and representation theory, Cambridge Tracts in Mathematics, vol 112. Cambridge University Press, Cambridge, reprint of the 1993 original Munn [1955] Munn WD (1955) On semigroup algebras. Proc Cambridge Philos Soc 51:1–15 Munn [1957a] Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Lawson MV (1998) Inverse semigroups. World Scientific Publishing Co. Inc., River Edge, NJ, 10.1142/9789812816689, URL http://dx.doi.org/10.1142/9789812816689, the theory of partial symmetries Martin [1994] Martin P (1994) Temperley-Lieb algebras for nonplanar statistical mechanics—the partition algebra construction. J Knot Theory Ramifications 3(1):51–82. 10.1142/S0218216594000071, URL http://dx.doi.org/10.1142/S0218216594000071 Martin [1996] Martin P (1996) The structure of the partition algebras. J Algebra 183(2):319–358. 10.1006/jabr.1996.0223, URL http://dx.doi.org/10.1006/jabr.1996.0223 Martin and Mazorchuk [2014] Martin P, Mazorchuk V (2014) On the representation theory of partial Brauer algebras. Q J Math 65(1):225–247. 10.1093/qmath/has043, URL https://doi.org/10.1093/qmath/has043 Martin [1990] Martin PP (1990) Representations of graph Temperley-Lieb algebras. Publ Res Inst Math Sci 26(3):485–503. 10.2977/prims/1195170958, URL https://doi.org/10.2977/prims/1195170958 Martin [2008] Martin S (2008) Schur algebras and representation theory, Cambridge Tracts in Mathematics, vol 112. Cambridge University Press, Cambridge, reprint of the 1993 original Munn [1955] Munn WD (1955) On semigroup algebras. Proc Cambridge Philos Soc 51:1–15 Munn [1957a] Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Martin P (1994) Temperley-Lieb algebras for nonplanar statistical mechanics—the partition algebra construction. J Knot Theory Ramifications 3(1):51–82. 10.1142/S0218216594000071, URL http://dx.doi.org/10.1142/S0218216594000071 Martin [1996] Martin P (1996) The structure of the partition algebras. J Algebra 183(2):319–358. 10.1006/jabr.1996.0223, URL http://dx.doi.org/10.1006/jabr.1996.0223 Martin and Mazorchuk [2014] Martin P, Mazorchuk V (2014) On the representation theory of partial Brauer algebras. Q J Math 65(1):225–247. 10.1093/qmath/has043, URL https://doi.org/10.1093/qmath/has043 Martin [1990] Martin PP (1990) Representations of graph Temperley-Lieb algebras. Publ Res Inst Math Sci 26(3):485–503. 10.2977/prims/1195170958, URL https://doi.org/10.2977/prims/1195170958 Martin [2008] Martin S (2008) Schur algebras and representation theory, Cambridge Tracts in Mathematics, vol 112. Cambridge University Press, Cambridge, reprint of the 1993 original Munn [1955] Munn WD (1955) On semigroup algebras. Proc Cambridge Philos Soc 51:1–15 Munn [1957a] Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Martin P (1996) The structure of the partition algebras. J Algebra 183(2):319–358. 10.1006/jabr.1996.0223, URL http://dx.doi.org/10.1006/jabr.1996.0223 Martin and Mazorchuk [2014] Martin P, Mazorchuk V (2014) On the representation theory of partial Brauer algebras. Q J Math 65(1):225–247. 10.1093/qmath/has043, URL https://doi.org/10.1093/qmath/has043 Martin [1990] Martin PP (1990) Representations of graph Temperley-Lieb algebras. Publ Res Inst Math Sci 26(3):485–503. 10.2977/prims/1195170958, URL https://doi.org/10.2977/prims/1195170958 Martin [2008] Martin S (2008) Schur algebras and representation theory, Cambridge Tracts in Mathematics, vol 112. Cambridge University Press, Cambridge, reprint of the 1993 original Munn [1955] Munn WD (1955) On semigroup algebras. Proc Cambridge Philos Soc 51:1–15 Munn [1957a] Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Martin P, Mazorchuk V (2014) On the representation theory of partial Brauer algebras. Q J Math 65(1):225–247. 10.1093/qmath/has043, URL https://doi.org/10.1093/qmath/has043 Martin [1990] Martin PP (1990) Representations of graph Temperley-Lieb algebras. Publ Res Inst Math Sci 26(3):485–503. 10.2977/prims/1195170958, URL https://doi.org/10.2977/prims/1195170958 Martin [2008] Martin S (2008) Schur algebras and representation theory, Cambridge Tracts in Mathematics, vol 112. Cambridge University Press, Cambridge, reprint of the 1993 original Munn [1955] Munn WD (1955) On semigroup algebras. Proc Cambridge Philos Soc 51:1–15 Munn [1957a] Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Martin PP (1990) Representations of graph Temperley-Lieb algebras. Publ Res Inst Math Sci 26(3):485–503. 10.2977/prims/1195170958, URL https://doi.org/10.2977/prims/1195170958 Martin [2008] Martin S (2008) Schur algebras and representation theory, Cambridge Tracts in Mathematics, vol 112. Cambridge University Press, Cambridge, reprint of the 1993 original Munn [1955] Munn WD (1955) On semigroup algebras. Proc Cambridge Philos Soc 51:1–15 Munn [1957a] Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Martin S (2008) Schur algebras and representation theory, Cambridge Tracts in Mathematics, vol 112. Cambridge University Press, Cambridge, reprint of the 1993 original Munn [1955] Munn WD (1955) On semigroup algebras. Proc Cambridge Philos Soc 51:1–15 Munn [1957a] Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Munn WD (1955) On semigroup algebras. Proc Cambridge Philos Soc 51:1–15 Munn [1957a] Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J.
- Grood C (2006) The rook partition algebra. J Combin Theory Ser A 113(2):325–351. 10.1016/j.jcta.2005.03.006, URL https://doi.org/10.1016/j.jcta.2005.03.006 Halverson [1996] Halverson T (1996) Characters of the centralizer algebras of mixed tensor representations of Gl(r,ℂ)Gl𝑟ℂ{\rm Gl}(r,\mathbb{C})roman_Gl ( italic_r , blackboard_C ) and the quantum group 𝔘q(ℊl(r,ℂ))subscript𝔘𝑞ℊ𝑙𝑟ℂ\mathfrak{U}_{q}({\mathcal{g}l}(r,\mathbb{C}))fraktur_U start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ( caligraphic_g italic_l ( italic_r , blackboard_C ) ). Pacific J Math 174(2):359–410. URL http://projecteuclid.org/euclid.pjm/1102365176 Halverson [2004] Halverson T (2004) Representations of the q𝑞qitalic_q-rook monoid. J Algebra 273(1):227–251. 10.1016/j.jalgebra.2003.11.002, URL http://dx.doi.org/10.1016/j.jalgebra.2003.11.002 Halverson and delMas [2014] Halverson T, delMas E (2014) Representations of the Rook-Brauer algebra. Comm Algebra 42(1):423–443. 10.1080/00927872.2012.716120, URL https://doi.org/10.1080/00927872.2012.716120 Halverson and Ram [2005] Halverson T, Ram A (2005) Partition algebras. European J Combin 26(6):869–921. 10.1016/j.ejc.2004.06.005, URL http://dx.doi.org/10.1016/j.ejc.2004.06.005 Jacobson [1989] Jacobson N (1989) Basic algebra. II, 2nd edn. W. H. Freeman and Company, New York Jones [1994] Jones VFR (1994) The Potts model and the symmetric group. In: Subfactors (Kyuzeso, 1993). World Sci. Publ., River Edge, NJ, p 259–267 Kudryavtseva and Mazorchuk [2008] Kudryavtseva G, Mazorchuk V (2008) Schur-Weyl dualities for symmetric inverse semigroups. J Pure Appl Algebra 212(8):1987–1995. 10.1016/j.jpaa.2007.12.004, URL https://doi.org/10.1016/j.jpaa.2007.12.004 Lawson [1998] Lawson MV (1998) Inverse semigroups. World Scientific Publishing Co. Inc., River Edge, NJ, 10.1142/9789812816689, URL http://dx.doi.org/10.1142/9789812816689, the theory of partial symmetries Martin [1994] Martin P (1994) Temperley-Lieb algebras for nonplanar statistical mechanics—the partition algebra construction. J Knot Theory Ramifications 3(1):51–82. 10.1142/S0218216594000071, URL http://dx.doi.org/10.1142/S0218216594000071 Martin [1996] Martin P (1996) The structure of the partition algebras. J Algebra 183(2):319–358. 10.1006/jabr.1996.0223, URL http://dx.doi.org/10.1006/jabr.1996.0223 Martin and Mazorchuk [2014] Martin P, Mazorchuk V (2014) On the representation theory of partial Brauer algebras. Q J Math 65(1):225–247. 10.1093/qmath/has043, URL https://doi.org/10.1093/qmath/has043 Martin [1990] Martin PP (1990) Representations of graph Temperley-Lieb algebras. Publ Res Inst Math Sci 26(3):485–503. 10.2977/prims/1195170958, URL https://doi.org/10.2977/prims/1195170958 Martin [2008] Martin S (2008) Schur algebras and representation theory, Cambridge Tracts in Mathematics, vol 112. Cambridge University Press, Cambridge, reprint of the 1993 original Munn [1955] Munn WD (1955) On semigroup algebras. Proc Cambridge Philos Soc 51:1–15 Munn [1957a] Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Halverson T (1996) Characters of the centralizer algebras of mixed tensor representations of Gl(r,ℂ)Gl𝑟ℂ{\rm Gl}(r,\mathbb{C})roman_Gl ( italic_r , blackboard_C ) and the quantum group 𝔘q(ℊl(r,ℂ))subscript𝔘𝑞ℊ𝑙𝑟ℂ\mathfrak{U}_{q}({\mathcal{g}l}(r,\mathbb{C}))fraktur_U start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ( caligraphic_g italic_l ( italic_r , blackboard_C ) ). Pacific J Math 174(2):359–410. URL http://projecteuclid.org/euclid.pjm/1102365176 Halverson [2004] Halverson T (2004) Representations of the q𝑞qitalic_q-rook monoid. J Algebra 273(1):227–251. 10.1016/j.jalgebra.2003.11.002, URL http://dx.doi.org/10.1016/j.jalgebra.2003.11.002 Halverson and delMas [2014] Halverson T, delMas E (2014) Representations of the Rook-Brauer algebra. Comm Algebra 42(1):423–443. 10.1080/00927872.2012.716120, URL https://doi.org/10.1080/00927872.2012.716120 Halverson and Ram [2005] Halverson T, Ram A (2005) Partition algebras. European J Combin 26(6):869–921. 10.1016/j.ejc.2004.06.005, URL http://dx.doi.org/10.1016/j.ejc.2004.06.005 Jacobson [1989] Jacobson N (1989) Basic algebra. II, 2nd edn. W. H. Freeman and Company, New York Jones [1994] Jones VFR (1994) The Potts model and the symmetric group. In: Subfactors (Kyuzeso, 1993). World Sci. Publ., River Edge, NJ, p 259–267 Kudryavtseva and Mazorchuk [2008] Kudryavtseva G, Mazorchuk V (2008) Schur-Weyl dualities for symmetric inverse semigroups. J Pure Appl Algebra 212(8):1987–1995. 10.1016/j.jpaa.2007.12.004, URL https://doi.org/10.1016/j.jpaa.2007.12.004 Lawson [1998] Lawson MV (1998) Inverse semigroups. World Scientific Publishing Co. Inc., River Edge, NJ, 10.1142/9789812816689, URL http://dx.doi.org/10.1142/9789812816689, the theory of partial symmetries Martin [1994] Martin P (1994) Temperley-Lieb algebras for nonplanar statistical mechanics—the partition algebra construction. J Knot Theory Ramifications 3(1):51–82. 10.1142/S0218216594000071, URL http://dx.doi.org/10.1142/S0218216594000071 Martin [1996] Martin P (1996) The structure of the partition algebras. J Algebra 183(2):319–358. 10.1006/jabr.1996.0223, URL http://dx.doi.org/10.1006/jabr.1996.0223 Martin and Mazorchuk [2014] Martin P, Mazorchuk V (2014) On the representation theory of partial Brauer algebras. Q J Math 65(1):225–247. 10.1093/qmath/has043, URL https://doi.org/10.1093/qmath/has043 Martin [1990] Martin PP (1990) Representations of graph Temperley-Lieb algebras. Publ Res Inst Math Sci 26(3):485–503. 10.2977/prims/1195170958, URL https://doi.org/10.2977/prims/1195170958 Martin [2008] Martin S (2008) Schur algebras and representation theory, Cambridge Tracts in Mathematics, vol 112. Cambridge University Press, Cambridge, reprint of the 1993 original Munn [1955] Munn WD (1955) On semigroup algebras. Proc Cambridge Philos Soc 51:1–15 Munn [1957a] Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Halverson T (2004) Representations of the q𝑞qitalic_q-rook monoid. J Algebra 273(1):227–251. 10.1016/j.jalgebra.2003.11.002, URL http://dx.doi.org/10.1016/j.jalgebra.2003.11.002 Halverson and delMas [2014] Halverson T, delMas E (2014) Representations of the Rook-Brauer algebra. Comm Algebra 42(1):423–443. 10.1080/00927872.2012.716120, URL https://doi.org/10.1080/00927872.2012.716120 Halverson and Ram [2005] Halverson T, Ram A (2005) Partition algebras. European J Combin 26(6):869–921. 10.1016/j.ejc.2004.06.005, URL http://dx.doi.org/10.1016/j.ejc.2004.06.005 Jacobson [1989] Jacobson N (1989) Basic algebra. II, 2nd edn. W. H. Freeman and Company, New York Jones [1994] Jones VFR (1994) The Potts model and the symmetric group. In: Subfactors (Kyuzeso, 1993). World Sci. Publ., River Edge, NJ, p 259–267 Kudryavtseva and Mazorchuk [2008] Kudryavtseva G, Mazorchuk V (2008) Schur-Weyl dualities for symmetric inverse semigroups. J Pure Appl Algebra 212(8):1987–1995. 10.1016/j.jpaa.2007.12.004, URL https://doi.org/10.1016/j.jpaa.2007.12.004 Lawson [1998] Lawson MV (1998) Inverse semigroups. World Scientific Publishing Co. Inc., River Edge, NJ, 10.1142/9789812816689, URL http://dx.doi.org/10.1142/9789812816689, the theory of partial symmetries Martin [1994] Martin P (1994) Temperley-Lieb algebras for nonplanar statistical mechanics—the partition algebra construction. J Knot Theory Ramifications 3(1):51–82. 10.1142/S0218216594000071, URL http://dx.doi.org/10.1142/S0218216594000071 Martin [1996] Martin P (1996) The structure of the partition algebras. J Algebra 183(2):319–358. 10.1006/jabr.1996.0223, URL http://dx.doi.org/10.1006/jabr.1996.0223 Martin and Mazorchuk [2014] Martin P, Mazorchuk V (2014) On the representation theory of partial Brauer algebras. Q J Math 65(1):225–247. 10.1093/qmath/has043, URL https://doi.org/10.1093/qmath/has043 Martin [1990] Martin PP (1990) Representations of graph Temperley-Lieb algebras. Publ Res Inst Math Sci 26(3):485–503. 10.2977/prims/1195170958, URL https://doi.org/10.2977/prims/1195170958 Martin [2008] Martin S (2008) Schur algebras and representation theory, Cambridge Tracts in Mathematics, vol 112. Cambridge University Press, Cambridge, reprint of the 1993 original Munn [1955] Munn WD (1955) On semigroup algebras. Proc Cambridge Philos Soc 51:1–15 Munn [1957a] Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Halverson T, delMas E (2014) Representations of the Rook-Brauer algebra. Comm Algebra 42(1):423–443. 10.1080/00927872.2012.716120, URL https://doi.org/10.1080/00927872.2012.716120 Halverson and Ram [2005] Halverson T, Ram A (2005) Partition algebras. European J Combin 26(6):869–921. 10.1016/j.ejc.2004.06.005, URL http://dx.doi.org/10.1016/j.ejc.2004.06.005 Jacobson [1989] Jacobson N (1989) Basic algebra. II, 2nd edn. W. H. Freeman and Company, New York Jones [1994] Jones VFR (1994) The Potts model and the symmetric group. In: Subfactors (Kyuzeso, 1993). World Sci. Publ., River Edge, NJ, p 259–267 Kudryavtseva and Mazorchuk [2008] Kudryavtseva G, Mazorchuk V (2008) Schur-Weyl dualities for symmetric inverse semigroups. J Pure Appl Algebra 212(8):1987–1995. 10.1016/j.jpaa.2007.12.004, URL https://doi.org/10.1016/j.jpaa.2007.12.004 Lawson [1998] Lawson MV (1998) Inverse semigroups. World Scientific Publishing Co. Inc., River Edge, NJ, 10.1142/9789812816689, URL http://dx.doi.org/10.1142/9789812816689, the theory of partial symmetries Martin [1994] Martin P (1994) Temperley-Lieb algebras for nonplanar statistical mechanics—the partition algebra construction. J Knot Theory Ramifications 3(1):51–82. 10.1142/S0218216594000071, URL http://dx.doi.org/10.1142/S0218216594000071 Martin [1996] Martin P (1996) The structure of the partition algebras. J Algebra 183(2):319–358. 10.1006/jabr.1996.0223, URL http://dx.doi.org/10.1006/jabr.1996.0223 Martin and Mazorchuk [2014] Martin P, Mazorchuk V (2014) On the representation theory of partial Brauer algebras. Q J Math 65(1):225–247. 10.1093/qmath/has043, URL https://doi.org/10.1093/qmath/has043 Martin [1990] Martin PP (1990) Representations of graph Temperley-Lieb algebras. Publ Res Inst Math Sci 26(3):485–503. 10.2977/prims/1195170958, URL https://doi.org/10.2977/prims/1195170958 Martin [2008] Martin S (2008) Schur algebras and representation theory, Cambridge Tracts in Mathematics, vol 112. Cambridge University Press, Cambridge, reprint of the 1993 original Munn [1955] Munn WD (1955) On semigroup algebras. Proc Cambridge Philos Soc 51:1–15 Munn [1957a] Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Halverson T, Ram A (2005) Partition algebras. European J Combin 26(6):869–921. 10.1016/j.ejc.2004.06.005, URL http://dx.doi.org/10.1016/j.ejc.2004.06.005 Jacobson [1989] Jacobson N (1989) Basic algebra. II, 2nd edn. W. H. Freeman and Company, New York Jones [1994] Jones VFR (1994) The Potts model and the symmetric group. In: Subfactors (Kyuzeso, 1993). World Sci. Publ., River Edge, NJ, p 259–267 Kudryavtseva and Mazorchuk [2008] Kudryavtseva G, Mazorchuk V (2008) Schur-Weyl dualities for symmetric inverse semigroups. J Pure Appl Algebra 212(8):1987–1995. 10.1016/j.jpaa.2007.12.004, URL https://doi.org/10.1016/j.jpaa.2007.12.004 Lawson [1998] Lawson MV (1998) Inverse semigroups. World Scientific Publishing Co. Inc., River Edge, NJ, 10.1142/9789812816689, URL http://dx.doi.org/10.1142/9789812816689, the theory of partial symmetries Martin [1994] Martin P (1994) Temperley-Lieb algebras for nonplanar statistical mechanics—the partition algebra construction. J Knot Theory Ramifications 3(1):51–82. 10.1142/S0218216594000071, URL http://dx.doi.org/10.1142/S0218216594000071 Martin [1996] Martin P (1996) The structure of the partition algebras. J Algebra 183(2):319–358. 10.1006/jabr.1996.0223, URL http://dx.doi.org/10.1006/jabr.1996.0223 Martin and Mazorchuk [2014] Martin P, Mazorchuk V (2014) On the representation theory of partial Brauer algebras. Q J Math 65(1):225–247. 10.1093/qmath/has043, URL https://doi.org/10.1093/qmath/has043 Martin [1990] Martin PP (1990) Representations of graph Temperley-Lieb algebras. Publ Res Inst Math Sci 26(3):485–503. 10.2977/prims/1195170958, URL https://doi.org/10.2977/prims/1195170958 Martin [2008] Martin S (2008) Schur algebras and representation theory, Cambridge Tracts in Mathematics, vol 112. Cambridge University Press, Cambridge, reprint of the 1993 original Munn [1955] Munn WD (1955) On semigroup algebras. Proc Cambridge Philos Soc 51:1–15 Munn [1957a] Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Jacobson N (1989) Basic algebra. II, 2nd edn. W. H. Freeman and Company, New York Jones [1994] Jones VFR (1994) The Potts model and the symmetric group. In: Subfactors (Kyuzeso, 1993). World Sci. Publ., River Edge, NJ, p 259–267 Kudryavtseva and Mazorchuk [2008] Kudryavtseva G, Mazorchuk V (2008) Schur-Weyl dualities for symmetric inverse semigroups. J Pure Appl Algebra 212(8):1987–1995. 10.1016/j.jpaa.2007.12.004, URL https://doi.org/10.1016/j.jpaa.2007.12.004 Lawson [1998] Lawson MV (1998) Inverse semigroups. World Scientific Publishing Co. Inc., River Edge, NJ, 10.1142/9789812816689, URL http://dx.doi.org/10.1142/9789812816689, the theory of partial symmetries Martin [1994] Martin P (1994) Temperley-Lieb algebras for nonplanar statistical mechanics—the partition algebra construction. J Knot Theory Ramifications 3(1):51–82. 10.1142/S0218216594000071, URL http://dx.doi.org/10.1142/S0218216594000071 Martin [1996] Martin P (1996) The structure of the partition algebras. J Algebra 183(2):319–358. 10.1006/jabr.1996.0223, URL http://dx.doi.org/10.1006/jabr.1996.0223 Martin and Mazorchuk [2014] Martin P, Mazorchuk V (2014) On the representation theory of partial Brauer algebras. Q J Math 65(1):225–247. 10.1093/qmath/has043, URL https://doi.org/10.1093/qmath/has043 Martin [1990] Martin PP (1990) Representations of graph Temperley-Lieb algebras. Publ Res Inst Math Sci 26(3):485–503. 10.2977/prims/1195170958, URL https://doi.org/10.2977/prims/1195170958 Martin [2008] Martin S (2008) Schur algebras and representation theory, Cambridge Tracts in Mathematics, vol 112. Cambridge University Press, Cambridge, reprint of the 1993 original Munn [1955] Munn WD (1955) On semigroup algebras. Proc Cambridge Philos Soc 51:1–15 Munn [1957a] Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Jones VFR (1994) The Potts model and the symmetric group. In: Subfactors (Kyuzeso, 1993). World Sci. Publ., River Edge, NJ, p 259–267 Kudryavtseva and Mazorchuk [2008] Kudryavtseva G, Mazorchuk V (2008) Schur-Weyl dualities for symmetric inverse semigroups. J Pure Appl Algebra 212(8):1987–1995. 10.1016/j.jpaa.2007.12.004, URL https://doi.org/10.1016/j.jpaa.2007.12.004 Lawson [1998] Lawson MV (1998) Inverse semigroups. World Scientific Publishing Co. Inc., River Edge, NJ, 10.1142/9789812816689, URL http://dx.doi.org/10.1142/9789812816689, the theory of partial symmetries Martin [1994] Martin P (1994) Temperley-Lieb algebras for nonplanar statistical mechanics—the partition algebra construction. J Knot Theory Ramifications 3(1):51–82. 10.1142/S0218216594000071, URL http://dx.doi.org/10.1142/S0218216594000071 Martin [1996] Martin P (1996) The structure of the partition algebras. J Algebra 183(2):319–358. 10.1006/jabr.1996.0223, URL http://dx.doi.org/10.1006/jabr.1996.0223 Martin and Mazorchuk [2014] Martin P, Mazorchuk V (2014) On the representation theory of partial Brauer algebras. Q J Math 65(1):225–247. 10.1093/qmath/has043, URL https://doi.org/10.1093/qmath/has043 Martin [1990] Martin PP (1990) Representations of graph Temperley-Lieb algebras. Publ Res Inst Math Sci 26(3):485–503. 10.2977/prims/1195170958, URL https://doi.org/10.2977/prims/1195170958 Martin [2008] Martin S (2008) Schur algebras and representation theory, Cambridge Tracts in Mathematics, vol 112. Cambridge University Press, Cambridge, reprint of the 1993 original Munn [1955] Munn WD (1955) On semigroup algebras. Proc Cambridge Philos Soc 51:1–15 Munn [1957a] Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Kudryavtseva G, Mazorchuk V (2008) Schur-Weyl dualities for symmetric inverse semigroups. J Pure Appl Algebra 212(8):1987–1995. 10.1016/j.jpaa.2007.12.004, URL https://doi.org/10.1016/j.jpaa.2007.12.004 Lawson [1998] Lawson MV (1998) Inverse semigroups. World Scientific Publishing Co. Inc., River Edge, NJ, 10.1142/9789812816689, URL http://dx.doi.org/10.1142/9789812816689, the theory of partial symmetries Martin [1994] Martin P (1994) Temperley-Lieb algebras for nonplanar statistical mechanics—the partition algebra construction. J Knot Theory Ramifications 3(1):51–82. 10.1142/S0218216594000071, URL http://dx.doi.org/10.1142/S0218216594000071 Martin [1996] Martin P (1996) The structure of the partition algebras. J Algebra 183(2):319–358. 10.1006/jabr.1996.0223, URL http://dx.doi.org/10.1006/jabr.1996.0223 Martin and Mazorchuk [2014] Martin P, Mazorchuk V (2014) On the representation theory of partial Brauer algebras. Q J Math 65(1):225–247. 10.1093/qmath/has043, URL https://doi.org/10.1093/qmath/has043 Martin [1990] Martin PP (1990) Representations of graph Temperley-Lieb algebras. Publ Res Inst Math Sci 26(3):485–503. 10.2977/prims/1195170958, URL https://doi.org/10.2977/prims/1195170958 Martin [2008] Martin S (2008) Schur algebras and representation theory, Cambridge Tracts in Mathematics, vol 112. Cambridge University Press, Cambridge, reprint of the 1993 original Munn [1955] Munn WD (1955) On semigroup algebras. Proc Cambridge Philos Soc 51:1–15 Munn [1957a] Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Lawson MV (1998) Inverse semigroups. World Scientific Publishing Co. Inc., River Edge, NJ, 10.1142/9789812816689, URL http://dx.doi.org/10.1142/9789812816689, the theory of partial symmetries Martin [1994] Martin P (1994) Temperley-Lieb algebras for nonplanar statistical mechanics—the partition algebra construction. J Knot Theory Ramifications 3(1):51–82. 10.1142/S0218216594000071, URL http://dx.doi.org/10.1142/S0218216594000071 Martin [1996] Martin P (1996) The structure of the partition algebras. J Algebra 183(2):319–358. 10.1006/jabr.1996.0223, URL http://dx.doi.org/10.1006/jabr.1996.0223 Martin and Mazorchuk [2014] Martin P, Mazorchuk V (2014) On the representation theory of partial Brauer algebras. Q J Math 65(1):225–247. 10.1093/qmath/has043, URL https://doi.org/10.1093/qmath/has043 Martin [1990] Martin PP (1990) Representations of graph Temperley-Lieb algebras. Publ Res Inst Math Sci 26(3):485–503. 10.2977/prims/1195170958, URL https://doi.org/10.2977/prims/1195170958 Martin [2008] Martin S (2008) Schur algebras and representation theory, Cambridge Tracts in Mathematics, vol 112. Cambridge University Press, Cambridge, reprint of the 1993 original Munn [1955] Munn WD (1955) On semigroup algebras. Proc Cambridge Philos Soc 51:1–15 Munn [1957a] Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Martin P (1994) Temperley-Lieb algebras for nonplanar statistical mechanics—the partition algebra construction. J Knot Theory Ramifications 3(1):51–82. 10.1142/S0218216594000071, URL http://dx.doi.org/10.1142/S0218216594000071 Martin [1996] Martin P (1996) The structure of the partition algebras. J Algebra 183(2):319–358. 10.1006/jabr.1996.0223, URL http://dx.doi.org/10.1006/jabr.1996.0223 Martin and Mazorchuk [2014] Martin P, Mazorchuk V (2014) On the representation theory of partial Brauer algebras. Q J Math 65(1):225–247. 10.1093/qmath/has043, URL https://doi.org/10.1093/qmath/has043 Martin [1990] Martin PP (1990) Representations of graph Temperley-Lieb algebras. Publ Res Inst Math Sci 26(3):485–503. 10.2977/prims/1195170958, URL https://doi.org/10.2977/prims/1195170958 Martin [2008] Martin S (2008) Schur algebras and representation theory, Cambridge Tracts in Mathematics, vol 112. Cambridge University Press, Cambridge, reprint of the 1993 original Munn [1955] Munn WD (1955) On semigroup algebras. Proc Cambridge Philos Soc 51:1–15 Munn [1957a] Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Martin P (1996) The structure of the partition algebras. J Algebra 183(2):319–358. 10.1006/jabr.1996.0223, URL http://dx.doi.org/10.1006/jabr.1996.0223 Martin and Mazorchuk [2014] Martin P, Mazorchuk V (2014) On the representation theory of partial Brauer algebras. Q J Math 65(1):225–247. 10.1093/qmath/has043, URL https://doi.org/10.1093/qmath/has043 Martin [1990] Martin PP (1990) Representations of graph Temperley-Lieb algebras. Publ Res Inst Math Sci 26(3):485–503. 10.2977/prims/1195170958, URL https://doi.org/10.2977/prims/1195170958 Martin [2008] Martin S (2008) Schur algebras and representation theory, Cambridge Tracts in Mathematics, vol 112. Cambridge University Press, Cambridge, reprint of the 1993 original Munn [1955] Munn WD (1955) On semigroup algebras. Proc Cambridge Philos Soc 51:1–15 Munn [1957a] Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Martin P, Mazorchuk V (2014) On the representation theory of partial Brauer algebras. Q J Math 65(1):225–247. 10.1093/qmath/has043, URL https://doi.org/10.1093/qmath/has043 Martin [1990] Martin PP (1990) Representations of graph Temperley-Lieb algebras. Publ Res Inst Math Sci 26(3):485–503. 10.2977/prims/1195170958, URL https://doi.org/10.2977/prims/1195170958 Martin [2008] Martin S (2008) Schur algebras and representation theory, Cambridge Tracts in Mathematics, vol 112. Cambridge University Press, Cambridge, reprint of the 1993 original Munn [1955] Munn WD (1955) On semigroup algebras. Proc Cambridge Philos Soc 51:1–15 Munn [1957a] Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Martin PP (1990) Representations of graph Temperley-Lieb algebras. Publ Res Inst Math Sci 26(3):485–503. 10.2977/prims/1195170958, URL https://doi.org/10.2977/prims/1195170958 Martin [2008] Martin S (2008) Schur algebras and representation theory, Cambridge Tracts in Mathematics, vol 112. Cambridge University Press, Cambridge, reprint of the 1993 original Munn [1955] Munn WD (1955) On semigroup algebras. Proc Cambridge Philos Soc 51:1–15 Munn [1957a] Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Martin S (2008) Schur algebras and representation theory, Cambridge Tracts in Mathematics, vol 112. Cambridge University Press, Cambridge, reprint of the 1993 original Munn [1955] Munn WD (1955) On semigroup algebras. Proc Cambridge Philos Soc 51:1–15 Munn [1957a] Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Munn WD (1955) On semigroup algebras. Proc Cambridge Philos Soc 51:1–15 Munn [1957a] Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J.
- Halverson T (1996) Characters of the centralizer algebras of mixed tensor representations of Gl(r,ℂ)Gl𝑟ℂ{\rm Gl}(r,\mathbb{C})roman_Gl ( italic_r , blackboard_C ) and the quantum group 𝔘q(ℊl(r,ℂ))subscript𝔘𝑞ℊ𝑙𝑟ℂ\mathfrak{U}_{q}({\mathcal{g}l}(r,\mathbb{C}))fraktur_U start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ( caligraphic_g italic_l ( italic_r , blackboard_C ) ). Pacific J Math 174(2):359–410. URL http://projecteuclid.org/euclid.pjm/1102365176 Halverson [2004] Halverson T (2004) Representations of the q𝑞qitalic_q-rook monoid. J Algebra 273(1):227–251. 10.1016/j.jalgebra.2003.11.002, URL http://dx.doi.org/10.1016/j.jalgebra.2003.11.002 Halverson and delMas [2014] Halverson T, delMas E (2014) Representations of the Rook-Brauer algebra. Comm Algebra 42(1):423–443. 10.1080/00927872.2012.716120, URL https://doi.org/10.1080/00927872.2012.716120 Halverson and Ram [2005] Halverson T, Ram A (2005) Partition algebras. European J Combin 26(6):869–921. 10.1016/j.ejc.2004.06.005, URL http://dx.doi.org/10.1016/j.ejc.2004.06.005 Jacobson [1989] Jacobson N (1989) Basic algebra. II, 2nd edn. W. H. Freeman and Company, New York Jones [1994] Jones VFR (1994) The Potts model and the symmetric group. In: Subfactors (Kyuzeso, 1993). World Sci. Publ., River Edge, NJ, p 259–267 Kudryavtseva and Mazorchuk [2008] Kudryavtseva G, Mazorchuk V (2008) Schur-Weyl dualities for symmetric inverse semigroups. J Pure Appl Algebra 212(8):1987–1995. 10.1016/j.jpaa.2007.12.004, URL https://doi.org/10.1016/j.jpaa.2007.12.004 Lawson [1998] Lawson MV (1998) Inverse semigroups. World Scientific Publishing Co. Inc., River Edge, NJ, 10.1142/9789812816689, URL http://dx.doi.org/10.1142/9789812816689, the theory of partial symmetries Martin [1994] Martin P (1994) Temperley-Lieb algebras for nonplanar statistical mechanics—the partition algebra construction. J Knot Theory Ramifications 3(1):51–82. 10.1142/S0218216594000071, URL http://dx.doi.org/10.1142/S0218216594000071 Martin [1996] Martin P (1996) The structure of the partition algebras. J Algebra 183(2):319–358. 10.1006/jabr.1996.0223, URL http://dx.doi.org/10.1006/jabr.1996.0223 Martin and Mazorchuk [2014] Martin P, Mazorchuk V (2014) On the representation theory of partial Brauer algebras. Q J Math 65(1):225–247. 10.1093/qmath/has043, URL https://doi.org/10.1093/qmath/has043 Martin [1990] Martin PP (1990) Representations of graph Temperley-Lieb algebras. Publ Res Inst Math Sci 26(3):485–503. 10.2977/prims/1195170958, URL https://doi.org/10.2977/prims/1195170958 Martin [2008] Martin S (2008) Schur algebras and representation theory, Cambridge Tracts in Mathematics, vol 112. Cambridge University Press, Cambridge, reprint of the 1993 original Munn [1955] Munn WD (1955) On semigroup algebras. Proc Cambridge Philos Soc 51:1–15 Munn [1957a] Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Halverson T (2004) Representations of the q𝑞qitalic_q-rook monoid. J Algebra 273(1):227–251. 10.1016/j.jalgebra.2003.11.002, URL http://dx.doi.org/10.1016/j.jalgebra.2003.11.002 Halverson and delMas [2014] Halverson T, delMas E (2014) Representations of the Rook-Brauer algebra. Comm Algebra 42(1):423–443. 10.1080/00927872.2012.716120, URL https://doi.org/10.1080/00927872.2012.716120 Halverson and Ram [2005] Halverson T, Ram A (2005) Partition algebras. European J Combin 26(6):869–921. 10.1016/j.ejc.2004.06.005, URL http://dx.doi.org/10.1016/j.ejc.2004.06.005 Jacobson [1989] Jacobson N (1989) Basic algebra. II, 2nd edn. W. H. Freeman and Company, New York Jones [1994] Jones VFR (1994) The Potts model and the symmetric group. In: Subfactors (Kyuzeso, 1993). World Sci. Publ., River Edge, NJ, p 259–267 Kudryavtseva and Mazorchuk [2008] Kudryavtseva G, Mazorchuk V (2008) Schur-Weyl dualities for symmetric inverse semigroups. J Pure Appl Algebra 212(8):1987–1995. 10.1016/j.jpaa.2007.12.004, URL https://doi.org/10.1016/j.jpaa.2007.12.004 Lawson [1998] Lawson MV (1998) Inverse semigroups. World Scientific Publishing Co. Inc., River Edge, NJ, 10.1142/9789812816689, URL http://dx.doi.org/10.1142/9789812816689, the theory of partial symmetries Martin [1994] Martin P (1994) Temperley-Lieb algebras for nonplanar statistical mechanics—the partition algebra construction. J Knot Theory Ramifications 3(1):51–82. 10.1142/S0218216594000071, URL http://dx.doi.org/10.1142/S0218216594000071 Martin [1996] Martin P (1996) The structure of the partition algebras. J Algebra 183(2):319–358. 10.1006/jabr.1996.0223, URL http://dx.doi.org/10.1006/jabr.1996.0223 Martin and Mazorchuk [2014] Martin P, Mazorchuk V (2014) On the representation theory of partial Brauer algebras. Q J Math 65(1):225–247. 10.1093/qmath/has043, URL https://doi.org/10.1093/qmath/has043 Martin [1990] Martin PP (1990) Representations of graph Temperley-Lieb algebras. Publ Res Inst Math Sci 26(3):485–503. 10.2977/prims/1195170958, URL https://doi.org/10.2977/prims/1195170958 Martin [2008] Martin S (2008) Schur algebras and representation theory, Cambridge Tracts in Mathematics, vol 112. Cambridge University Press, Cambridge, reprint of the 1993 original Munn [1955] Munn WD (1955) On semigroup algebras. Proc Cambridge Philos Soc 51:1–15 Munn [1957a] Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Halverson T, delMas E (2014) Representations of the Rook-Brauer algebra. Comm Algebra 42(1):423–443. 10.1080/00927872.2012.716120, URL https://doi.org/10.1080/00927872.2012.716120 Halverson and Ram [2005] Halverson T, Ram A (2005) Partition algebras. European J Combin 26(6):869–921. 10.1016/j.ejc.2004.06.005, URL http://dx.doi.org/10.1016/j.ejc.2004.06.005 Jacobson [1989] Jacobson N (1989) Basic algebra. II, 2nd edn. W. H. Freeman and Company, New York Jones [1994] Jones VFR (1994) The Potts model and the symmetric group. In: Subfactors (Kyuzeso, 1993). World Sci. Publ., River Edge, NJ, p 259–267 Kudryavtseva and Mazorchuk [2008] Kudryavtseva G, Mazorchuk V (2008) Schur-Weyl dualities for symmetric inverse semigroups. J Pure Appl Algebra 212(8):1987–1995. 10.1016/j.jpaa.2007.12.004, URL https://doi.org/10.1016/j.jpaa.2007.12.004 Lawson [1998] Lawson MV (1998) Inverse semigroups. World Scientific Publishing Co. Inc., River Edge, NJ, 10.1142/9789812816689, URL http://dx.doi.org/10.1142/9789812816689, the theory of partial symmetries Martin [1994] Martin P (1994) Temperley-Lieb algebras for nonplanar statistical mechanics—the partition algebra construction. J Knot Theory Ramifications 3(1):51–82. 10.1142/S0218216594000071, URL http://dx.doi.org/10.1142/S0218216594000071 Martin [1996] Martin P (1996) The structure of the partition algebras. J Algebra 183(2):319–358. 10.1006/jabr.1996.0223, URL http://dx.doi.org/10.1006/jabr.1996.0223 Martin and Mazorchuk [2014] Martin P, Mazorchuk V (2014) On the representation theory of partial Brauer algebras. Q J Math 65(1):225–247. 10.1093/qmath/has043, URL https://doi.org/10.1093/qmath/has043 Martin [1990] Martin PP (1990) Representations of graph Temperley-Lieb algebras. Publ Res Inst Math Sci 26(3):485–503. 10.2977/prims/1195170958, URL https://doi.org/10.2977/prims/1195170958 Martin [2008] Martin S (2008) Schur algebras and representation theory, Cambridge Tracts in Mathematics, vol 112. Cambridge University Press, Cambridge, reprint of the 1993 original Munn [1955] Munn WD (1955) On semigroup algebras. Proc Cambridge Philos Soc 51:1–15 Munn [1957a] Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Halverson T, Ram A (2005) Partition algebras. European J Combin 26(6):869–921. 10.1016/j.ejc.2004.06.005, URL http://dx.doi.org/10.1016/j.ejc.2004.06.005 Jacobson [1989] Jacobson N (1989) Basic algebra. II, 2nd edn. W. H. Freeman and Company, New York Jones [1994] Jones VFR (1994) The Potts model and the symmetric group. In: Subfactors (Kyuzeso, 1993). World Sci. Publ., River Edge, NJ, p 259–267 Kudryavtseva and Mazorchuk [2008] Kudryavtseva G, Mazorchuk V (2008) Schur-Weyl dualities for symmetric inverse semigroups. J Pure Appl Algebra 212(8):1987–1995. 10.1016/j.jpaa.2007.12.004, URL https://doi.org/10.1016/j.jpaa.2007.12.004 Lawson [1998] Lawson MV (1998) Inverse semigroups. World Scientific Publishing Co. Inc., River Edge, NJ, 10.1142/9789812816689, URL http://dx.doi.org/10.1142/9789812816689, the theory of partial symmetries Martin [1994] Martin P (1994) Temperley-Lieb algebras for nonplanar statistical mechanics—the partition algebra construction. J Knot Theory Ramifications 3(1):51–82. 10.1142/S0218216594000071, URL http://dx.doi.org/10.1142/S0218216594000071 Martin [1996] Martin P (1996) The structure of the partition algebras. J Algebra 183(2):319–358. 10.1006/jabr.1996.0223, URL http://dx.doi.org/10.1006/jabr.1996.0223 Martin and Mazorchuk [2014] Martin P, Mazorchuk V (2014) On the representation theory of partial Brauer algebras. Q J Math 65(1):225–247. 10.1093/qmath/has043, URL https://doi.org/10.1093/qmath/has043 Martin [1990] Martin PP (1990) Representations of graph Temperley-Lieb algebras. Publ Res Inst Math Sci 26(3):485–503. 10.2977/prims/1195170958, URL https://doi.org/10.2977/prims/1195170958 Martin [2008] Martin S (2008) Schur algebras and representation theory, Cambridge Tracts in Mathematics, vol 112. Cambridge University Press, Cambridge, reprint of the 1993 original Munn [1955] Munn WD (1955) On semigroup algebras. Proc Cambridge Philos Soc 51:1–15 Munn [1957a] Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Jacobson N (1989) Basic algebra. II, 2nd edn. W. H. Freeman and Company, New York Jones [1994] Jones VFR (1994) The Potts model and the symmetric group. In: Subfactors (Kyuzeso, 1993). World Sci. Publ., River Edge, NJ, p 259–267 Kudryavtseva and Mazorchuk [2008] Kudryavtseva G, Mazorchuk V (2008) Schur-Weyl dualities for symmetric inverse semigroups. J Pure Appl Algebra 212(8):1987–1995. 10.1016/j.jpaa.2007.12.004, URL https://doi.org/10.1016/j.jpaa.2007.12.004 Lawson [1998] Lawson MV (1998) Inverse semigroups. World Scientific Publishing Co. Inc., River Edge, NJ, 10.1142/9789812816689, URL http://dx.doi.org/10.1142/9789812816689, the theory of partial symmetries Martin [1994] Martin P (1994) Temperley-Lieb algebras for nonplanar statistical mechanics—the partition algebra construction. J Knot Theory Ramifications 3(1):51–82. 10.1142/S0218216594000071, URL http://dx.doi.org/10.1142/S0218216594000071 Martin [1996] Martin P (1996) The structure of the partition algebras. J Algebra 183(2):319–358. 10.1006/jabr.1996.0223, URL http://dx.doi.org/10.1006/jabr.1996.0223 Martin and Mazorchuk [2014] Martin P, Mazorchuk V (2014) On the representation theory of partial Brauer algebras. Q J Math 65(1):225–247. 10.1093/qmath/has043, URL https://doi.org/10.1093/qmath/has043 Martin [1990] Martin PP (1990) Representations of graph Temperley-Lieb algebras. Publ Res Inst Math Sci 26(3):485–503. 10.2977/prims/1195170958, URL https://doi.org/10.2977/prims/1195170958 Martin [2008] Martin S (2008) Schur algebras and representation theory, Cambridge Tracts in Mathematics, vol 112. Cambridge University Press, Cambridge, reprint of the 1993 original Munn [1955] Munn WD (1955) On semigroup algebras. Proc Cambridge Philos Soc 51:1–15 Munn [1957a] Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Jones VFR (1994) The Potts model and the symmetric group. In: Subfactors (Kyuzeso, 1993). World Sci. Publ., River Edge, NJ, p 259–267 Kudryavtseva and Mazorchuk [2008] Kudryavtseva G, Mazorchuk V (2008) Schur-Weyl dualities for symmetric inverse semigroups. J Pure Appl Algebra 212(8):1987–1995. 10.1016/j.jpaa.2007.12.004, URL https://doi.org/10.1016/j.jpaa.2007.12.004 Lawson [1998] Lawson MV (1998) Inverse semigroups. World Scientific Publishing Co. Inc., River Edge, NJ, 10.1142/9789812816689, URL http://dx.doi.org/10.1142/9789812816689, the theory of partial symmetries Martin [1994] Martin P (1994) Temperley-Lieb algebras for nonplanar statistical mechanics—the partition algebra construction. J Knot Theory Ramifications 3(1):51–82. 10.1142/S0218216594000071, URL http://dx.doi.org/10.1142/S0218216594000071 Martin [1996] Martin P (1996) The structure of the partition algebras. J Algebra 183(2):319–358. 10.1006/jabr.1996.0223, URL http://dx.doi.org/10.1006/jabr.1996.0223 Martin and Mazorchuk [2014] Martin P, Mazorchuk V (2014) On the representation theory of partial Brauer algebras. Q J Math 65(1):225–247. 10.1093/qmath/has043, URL https://doi.org/10.1093/qmath/has043 Martin [1990] Martin PP (1990) Representations of graph Temperley-Lieb algebras. Publ Res Inst Math Sci 26(3):485–503. 10.2977/prims/1195170958, URL https://doi.org/10.2977/prims/1195170958 Martin [2008] Martin S (2008) Schur algebras and representation theory, Cambridge Tracts in Mathematics, vol 112. Cambridge University Press, Cambridge, reprint of the 1993 original Munn [1955] Munn WD (1955) On semigroup algebras. Proc Cambridge Philos Soc 51:1–15 Munn [1957a] Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Kudryavtseva G, Mazorchuk V (2008) Schur-Weyl dualities for symmetric inverse semigroups. J Pure Appl Algebra 212(8):1987–1995. 10.1016/j.jpaa.2007.12.004, URL https://doi.org/10.1016/j.jpaa.2007.12.004 Lawson [1998] Lawson MV (1998) Inverse semigroups. World Scientific Publishing Co. Inc., River Edge, NJ, 10.1142/9789812816689, URL http://dx.doi.org/10.1142/9789812816689, the theory of partial symmetries Martin [1994] Martin P (1994) Temperley-Lieb algebras for nonplanar statistical mechanics—the partition algebra construction. J Knot Theory Ramifications 3(1):51–82. 10.1142/S0218216594000071, URL http://dx.doi.org/10.1142/S0218216594000071 Martin [1996] Martin P (1996) The structure of the partition algebras. J Algebra 183(2):319–358. 10.1006/jabr.1996.0223, URL http://dx.doi.org/10.1006/jabr.1996.0223 Martin and Mazorchuk [2014] Martin P, Mazorchuk V (2014) On the representation theory of partial Brauer algebras. Q J Math 65(1):225–247. 10.1093/qmath/has043, URL https://doi.org/10.1093/qmath/has043 Martin [1990] Martin PP (1990) Representations of graph Temperley-Lieb algebras. Publ Res Inst Math Sci 26(3):485–503. 10.2977/prims/1195170958, URL https://doi.org/10.2977/prims/1195170958 Martin [2008] Martin S (2008) Schur algebras and representation theory, Cambridge Tracts in Mathematics, vol 112. Cambridge University Press, Cambridge, reprint of the 1993 original Munn [1955] Munn WD (1955) On semigroup algebras. Proc Cambridge Philos Soc 51:1–15 Munn [1957a] Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Lawson MV (1998) Inverse semigroups. World Scientific Publishing Co. Inc., River Edge, NJ, 10.1142/9789812816689, URL http://dx.doi.org/10.1142/9789812816689, the theory of partial symmetries Martin [1994] Martin P (1994) Temperley-Lieb algebras for nonplanar statistical mechanics—the partition algebra construction. J Knot Theory Ramifications 3(1):51–82. 10.1142/S0218216594000071, URL http://dx.doi.org/10.1142/S0218216594000071 Martin [1996] Martin P (1996) The structure of the partition algebras. J Algebra 183(2):319–358. 10.1006/jabr.1996.0223, URL http://dx.doi.org/10.1006/jabr.1996.0223 Martin and Mazorchuk [2014] Martin P, Mazorchuk V (2014) On the representation theory of partial Brauer algebras. Q J Math 65(1):225–247. 10.1093/qmath/has043, URL https://doi.org/10.1093/qmath/has043 Martin [1990] Martin PP (1990) Representations of graph Temperley-Lieb algebras. Publ Res Inst Math Sci 26(3):485–503. 10.2977/prims/1195170958, URL https://doi.org/10.2977/prims/1195170958 Martin [2008] Martin S (2008) Schur algebras and representation theory, Cambridge Tracts in Mathematics, vol 112. Cambridge University Press, Cambridge, reprint of the 1993 original Munn [1955] Munn WD (1955) On semigroup algebras. Proc Cambridge Philos Soc 51:1–15 Munn [1957a] Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Martin P (1994) Temperley-Lieb algebras for nonplanar statistical mechanics—the partition algebra construction. J Knot Theory Ramifications 3(1):51–82. 10.1142/S0218216594000071, URL http://dx.doi.org/10.1142/S0218216594000071 Martin [1996] Martin P (1996) The structure of the partition algebras. J Algebra 183(2):319–358. 10.1006/jabr.1996.0223, URL http://dx.doi.org/10.1006/jabr.1996.0223 Martin and Mazorchuk [2014] Martin P, Mazorchuk V (2014) On the representation theory of partial Brauer algebras. Q J Math 65(1):225–247. 10.1093/qmath/has043, URL https://doi.org/10.1093/qmath/has043 Martin [1990] Martin PP (1990) Representations of graph Temperley-Lieb algebras. Publ Res Inst Math Sci 26(3):485–503. 10.2977/prims/1195170958, URL https://doi.org/10.2977/prims/1195170958 Martin [2008] Martin S (2008) Schur algebras and representation theory, Cambridge Tracts in Mathematics, vol 112. Cambridge University Press, Cambridge, reprint of the 1993 original Munn [1955] Munn WD (1955) On semigroup algebras. Proc Cambridge Philos Soc 51:1–15 Munn [1957a] Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Martin P (1996) The structure of the partition algebras. J Algebra 183(2):319–358. 10.1006/jabr.1996.0223, URL http://dx.doi.org/10.1006/jabr.1996.0223 Martin and Mazorchuk [2014] Martin P, Mazorchuk V (2014) On the representation theory of partial Brauer algebras. Q J Math 65(1):225–247. 10.1093/qmath/has043, URL https://doi.org/10.1093/qmath/has043 Martin [1990] Martin PP (1990) Representations of graph Temperley-Lieb algebras. Publ Res Inst Math Sci 26(3):485–503. 10.2977/prims/1195170958, URL https://doi.org/10.2977/prims/1195170958 Martin [2008] Martin S (2008) Schur algebras and representation theory, Cambridge Tracts in Mathematics, vol 112. Cambridge University Press, Cambridge, reprint of the 1993 original Munn [1955] Munn WD (1955) On semigroup algebras. Proc Cambridge Philos Soc 51:1–15 Munn [1957a] Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Martin P, Mazorchuk V (2014) On the representation theory of partial Brauer algebras. Q J Math 65(1):225–247. 10.1093/qmath/has043, URL https://doi.org/10.1093/qmath/has043 Martin [1990] Martin PP (1990) Representations of graph Temperley-Lieb algebras. Publ Res Inst Math Sci 26(3):485–503. 10.2977/prims/1195170958, URL https://doi.org/10.2977/prims/1195170958 Martin [2008] Martin S (2008) Schur algebras and representation theory, Cambridge Tracts in Mathematics, vol 112. Cambridge University Press, Cambridge, reprint of the 1993 original Munn [1955] Munn WD (1955) On semigroup algebras. Proc Cambridge Philos Soc 51:1–15 Munn [1957a] Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Martin PP (1990) Representations of graph Temperley-Lieb algebras. Publ Res Inst Math Sci 26(3):485–503. 10.2977/prims/1195170958, URL https://doi.org/10.2977/prims/1195170958 Martin [2008] Martin S (2008) Schur algebras and representation theory, Cambridge Tracts in Mathematics, vol 112. Cambridge University Press, Cambridge, reprint of the 1993 original Munn [1955] Munn WD (1955) On semigroup algebras. Proc Cambridge Philos Soc 51:1–15 Munn [1957a] Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Martin S (2008) Schur algebras and representation theory, Cambridge Tracts in Mathematics, vol 112. Cambridge University Press, Cambridge, reprint of the 1993 original Munn [1955] Munn WD (1955) On semigroup algebras. Proc Cambridge Philos Soc 51:1–15 Munn [1957a] Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Munn WD (1955) On semigroup algebras. Proc Cambridge Philos Soc 51:1–15 Munn [1957a] Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. 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Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Halverson T, delMas E (2014) Representations of the Rook-Brauer algebra. Comm Algebra 42(1):423–443. 10.1080/00927872.2012.716120, URL https://doi.org/10.1080/00927872.2012.716120 Halverson and Ram [2005] Halverson T, Ram A (2005) Partition algebras. European J Combin 26(6):869–921. 10.1016/j.ejc.2004.06.005, URL http://dx.doi.org/10.1016/j.ejc.2004.06.005 Jacobson [1989] Jacobson N (1989) Basic algebra. II, 2nd edn. W. H. Freeman and Company, New York Jones [1994] Jones VFR (1994) The Potts model and the symmetric group. In: Subfactors (Kyuzeso, 1993). World Sci. Publ., River Edge, NJ, p 259–267 Kudryavtseva and Mazorchuk [2008] Kudryavtseva G, Mazorchuk V (2008) Schur-Weyl dualities for symmetric inverse semigroups. J Pure Appl Algebra 212(8):1987–1995. 10.1016/j.jpaa.2007.12.004, URL https://doi.org/10.1016/j.jpaa.2007.12.004 Lawson [1998] Lawson MV (1998) Inverse semigroups. World Scientific Publishing Co. Inc., River Edge, NJ, 10.1142/9789812816689, URL http://dx.doi.org/10.1142/9789812816689, the theory of partial symmetries Martin [1994] Martin P (1994) Temperley-Lieb algebras for nonplanar statistical mechanics—the partition algebra construction. J Knot Theory Ramifications 3(1):51–82. 10.1142/S0218216594000071, URL http://dx.doi.org/10.1142/S0218216594000071 Martin [1996] Martin P (1996) The structure of the partition algebras. J Algebra 183(2):319–358. 10.1006/jabr.1996.0223, URL http://dx.doi.org/10.1006/jabr.1996.0223 Martin and Mazorchuk [2014] Martin P, Mazorchuk V (2014) On the representation theory of partial Brauer algebras. Q J Math 65(1):225–247. 10.1093/qmath/has043, URL https://doi.org/10.1093/qmath/has043 Martin [1990] Martin PP (1990) Representations of graph Temperley-Lieb algebras. Publ Res Inst Math Sci 26(3):485–503. 10.2977/prims/1195170958, URL https://doi.org/10.2977/prims/1195170958 Martin [2008] Martin S (2008) Schur algebras and representation theory, Cambridge Tracts in Mathematics, vol 112. Cambridge University Press, Cambridge, reprint of the 1993 original Munn [1955] Munn WD (1955) On semigroup algebras. Proc Cambridge Philos Soc 51:1–15 Munn [1957a] Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Halverson T, Ram A (2005) Partition algebras. European J Combin 26(6):869–921. 10.1016/j.ejc.2004.06.005, URL http://dx.doi.org/10.1016/j.ejc.2004.06.005 Jacobson [1989] Jacobson N (1989) Basic algebra. II, 2nd edn. W. H. Freeman and Company, New York Jones [1994] Jones VFR (1994) The Potts model and the symmetric group. In: Subfactors (Kyuzeso, 1993). World Sci. Publ., River Edge, NJ, p 259–267 Kudryavtseva and Mazorchuk [2008] Kudryavtseva G, Mazorchuk V (2008) Schur-Weyl dualities for symmetric inverse semigroups. J Pure Appl Algebra 212(8):1987–1995. 10.1016/j.jpaa.2007.12.004, URL https://doi.org/10.1016/j.jpaa.2007.12.004 Lawson [1998] Lawson MV (1998) Inverse semigroups. World Scientific Publishing Co. Inc., River Edge, NJ, 10.1142/9789812816689, URL http://dx.doi.org/10.1142/9789812816689, the theory of partial symmetries Martin [1994] Martin P (1994) Temperley-Lieb algebras for nonplanar statistical mechanics—the partition algebra construction. J Knot Theory Ramifications 3(1):51–82. 10.1142/S0218216594000071, URL http://dx.doi.org/10.1142/S0218216594000071 Martin [1996] Martin P (1996) The structure of the partition algebras. J Algebra 183(2):319–358. 10.1006/jabr.1996.0223, URL http://dx.doi.org/10.1006/jabr.1996.0223 Martin and Mazorchuk [2014] Martin P, Mazorchuk V (2014) On the representation theory of partial Brauer algebras. Q J Math 65(1):225–247. 10.1093/qmath/has043, URL https://doi.org/10.1093/qmath/has043 Martin [1990] Martin PP (1990) Representations of graph Temperley-Lieb algebras. Publ Res Inst Math Sci 26(3):485–503. 10.2977/prims/1195170958, URL https://doi.org/10.2977/prims/1195170958 Martin [2008] Martin S (2008) Schur algebras and representation theory, Cambridge Tracts in Mathematics, vol 112. Cambridge University Press, Cambridge, reprint of the 1993 original Munn [1955] Munn WD (1955) On semigroup algebras. Proc Cambridge Philos Soc 51:1–15 Munn [1957a] Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Jacobson N (1989) Basic algebra. II, 2nd edn. W. H. Freeman and Company, New York Jones [1994] Jones VFR (1994) The Potts model and the symmetric group. In: Subfactors (Kyuzeso, 1993). World Sci. Publ., River Edge, NJ, p 259–267 Kudryavtseva and Mazorchuk [2008] Kudryavtseva G, Mazorchuk V (2008) Schur-Weyl dualities for symmetric inverse semigroups. J Pure Appl Algebra 212(8):1987–1995. 10.1016/j.jpaa.2007.12.004, URL https://doi.org/10.1016/j.jpaa.2007.12.004 Lawson [1998] Lawson MV (1998) Inverse semigroups. World Scientific Publishing Co. Inc., River Edge, NJ, 10.1142/9789812816689, URL http://dx.doi.org/10.1142/9789812816689, the theory of partial symmetries Martin [1994] Martin P (1994) Temperley-Lieb algebras for nonplanar statistical mechanics—the partition algebra construction. J Knot Theory Ramifications 3(1):51–82. 10.1142/S0218216594000071, URL http://dx.doi.org/10.1142/S0218216594000071 Martin [1996] Martin P (1996) The structure of the partition algebras. J Algebra 183(2):319–358. 10.1006/jabr.1996.0223, URL http://dx.doi.org/10.1006/jabr.1996.0223 Martin and Mazorchuk [2014] Martin P, Mazorchuk V (2014) On the representation theory of partial Brauer algebras. Q J Math 65(1):225–247. 10.1093/qmath/has043, URL https://doi.org/10.1093/qmath/has043 Martin [1990] Martin PP (1990) Representations of graph Temperley-Lieb algebras. Publ Res Inst Math Sci 26(3):485–503. 10.2977/prims/1195170958, URL https://doi.org/10.2977/prims/1195170958 Martin [2008] Martin S (2008) Schur algebras and representation theory, Cambridge Tracts in Mathematics, vol 112. Cambridge University Press, Cambridge, reprint of the 1993 original Munn [1955] Munn WD (1955) On semigroup algebras. Proc Cambridge Philos Soc 51:1–15 Munn [1957a] Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Jones VFR (1994) The Potts model and the symmetric group. In: Subfactors (Kyuzeso, 1993). World Sci. Publ., River Edge, NJ, p 259–267 Kudryavtseva and Mazorchuk [2008] Kudryavtseva G, Mazorchuk V (2008) Schur-Weyl dualities for symmetric inverse semigroups. J Pure Appl Algebra 212(8):1987–1995. 10.1016/j.jpaa.2007.12.004, URL https://doi.org/10.1016/j.jpaa.2007.12.004 Lawson [1998] Lawson MV (1998) Inverse semigroups. World Scientific Publishing Co. Inc., River Edge, NJ, 10.1142/9789812816689, URL http://dx.doi.org/10.1142/9789812816689, the theory of partial symmetries Martin [1994] Martin P (1994) Temperley-Lieb algebras for nonplanar statistical mechanics—the partition algebra construction. J Knot Theory Ramifications 3(1):51–82. 10.1142/S0218216594000071, URL http://dx.doi.org/10.1142/S0218216594000071 Martin [1996] Martin P (1996) The structure of the partition algebras. J Algebra 183(2):319–358. 10.1006/jabr.1996.0223, URL http://dx.doi.org/10.1006/jabr.1996.0223 Martin and Mazorchuk [2014] Martin P, Mazorchuk V (2014) On the representation theory of partial Brauer algebras. Q J Math 65(1):225–247. 10.1093/qmath/has043, URL https://doi.org/10.1093/qmath/has043 Martin [1990] Martin PP (1990) Representations of graph Temperley-Lieb algebras. Publ Res Inst Math Sci 26(3):485–503. 10.2977/prims/1195170958, URL https://doi.org/10.2977/prims/1195170958 Martin [2008] Martin S (2008) Schur algebras and representation theory, Cambridge Tracts in Mathematics, vol 112. Cambridge University Press, Cambridge, reprint of the 1993 original Munn [1955] Munn WD (1955) On semigroup algebras. Proc Cambridge Philos Soc 51:1–15 Munn [1957a] Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Kudryavtseva G, Mazorchuk V (2008) Schur-Weyl dualities for symmetric inverse semigroups. J Pure Appl Algebra 212(8):1987–1995. 10.1016/j.jpaa.2007.12.004, URL https://doi.org/10.1016/j.jpaa.2007.12.004 Lawson [1998] Lawson MV (1998) Inverse semigroups. World Scientific Publishing Co. Inc., River Edge, NJ, 10.1142/9789812816689, URL http://dx.doi.org/10.1142/9789812816689, the theory of partial symmetries Martin [1994] Martin P (1994) Temperley-Lieb algebras for nonplanar statistical mechanics—the partition algebra construction. J Knot Theory Ramifications 3(1):51–82. 10.1142/S0218216594000071, URL http://dx.doi.org/10.1142/S0218216594000071 Martin [1996] Martin P (1996) The structure of the partition algebras. J Algebra 183(2):319–358. 10.1006/jabr.1996.0223, URL http://dx.doi.org/10.1006/jabr.1996.0223 Martin and Mazorchuk [2014] Martin P, Mazorchuk V (2014) On the representation theory of partial Brauer algebras. Q J Math 65(1):225–247. 10.1093/qmath/has043, URL https://doi.org/10.1093/qmath/has043 Martin [1990] Martin PP (1990) Representations of graph Temperley-Lieb algebras. Publ Res Inst Math Sci 26(3):485–503. 10.2977/prims/1195170958, URL https://doi.org/10.2977/prims/1195170958 Martin [2008] Martin S (2008) Schur algebras and representation theory, Cambridge Tracts in Mathematics, vol 112. Cambridge University Press, Cambridge, reprint of the 1993 original Munn [1955] Munn WD (1955) On semigroup algebras. Proc Cambridge Philos Soc 51:1–15 Munn [1957a] Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Lawson MV (1998) Inverse semigroups. World Scientific Publishing Co. Inc., River Edge, NJ, 10.1142/9789812816689, URL http://dx.doi.org/10.1142/9789812816689, the theory of partial symmetries Martin [1994] Martin P (1994) Temperley-Lieb algebras for nonplanar statistical mechanics—the partition algebra construction. J Knot Theory Ramifications 3(1):51–82. 10.1142/S0218216594000071, URL http://dx.doi.org/10.1142/S0218216594000071 Martin [1996] Martin P (1996) The structure of the partition algebras. J Algebra 183(2):319–358. 10.1006/jabr.1996.0223, URL http://dx.doi.org/10.1006/jabr.1996.0223 Martin and Mazorchuk [2014] Martin P, Mazorchuk V (2014) On the representation theory of partial Brauer algebras. Q J Math 65(1):225–247. 10.1093/qmath/has043, URL https://doi.org/10.1093/qmath/has043 Martin [1990] Martin PP (1990) Representations of graph Temperley-Lieb algebras. Publ Res Inst Math Sci 26(3):485–503. 10.2977/prims/1195170958, URL https://doi.org/10.2977/prims/1195170958 Martin [2008] Martin S (2008) Schur algebras and representation theory, Cambridge Tracts in Mathematics, vol 112. Cambridge University Press, Cambridge, reprint of the 1993 original Munn [1955] Munn WD (1955) On semigroup algebras. Proc Cambridge Philos Soc 51:1–15 Munn [1957a] Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Martin P (1994) Temperley-Lieb algebras for nonplanar statistical mechanics—the partition algebra construction. J Knot Theory Ramifications 3(1):51–82. 10.1142/S0218216594000071, URL http://dx.doi.org/10.1142/S0218216594000071 Martin [1996] Martin P (1996) The structure of the partition algebras. J Algebra 183(2):319–358. 10.1006/jabr.1996.0223, URL http://dx.doi.org/10.1006/jabr.1996.0223 Martin and Mazorchuk [2014] Martin P, Mazorchuk V (2014) On the representation theory of partial Brauer algebras. Q J Math 65(1):225–247. 10.1093/qmath/has043, URL https://doi.org/10.1093/qmath/has043 Martin [1990] Martin PP (1990) Representations of graph Temperley-Lieb algebras. Publ Res Inst Math Sci 26(3):485–503. 10.2977/prims/1195170958, URL https://doi.org/10.2977/prims/1195170958 Martin [2008] Martin S (2008) Schur algebras and representation theory, Cambridge Tracts in Mathematics, vol 112. Cambridge University Press, Cambridge, reprint of the 1993 original Munn [1955] Munn WD (1955) On semigroup algebras. Proc Cambridge Philos Soc 51:1–15 Munn [1957a] Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Martin P (1996) The structure of the partition algebras. J Algebra 183(2):319–358. 10.1006/jabr.1996.0223, URL http://dx.doi.org/10.1006/jabr.1996.0223 Martin and Mazorchuk [2014] Martin P, Mazorchuk V (2014) On the representation theory of partial Brauer algebras. Q J Math 65(1):225–247. 10.1093/qmath/has043, URL https://doi.org/10.1093/qmath/has043 Martin [1990] Martin PP (1990) Representations of graph Temperley-Lieb algebras. Publ Res Inst Math Sci 26(3):485–503. 10.2977/prims/1195170958, URL https://doi.org/10.2977/prims/1195170958 Martin [2008] Martin S (2008) Schur algebras and representation theory, Cambridge Tracts in Mathematics, vol 112. Cambridge University Press, Cambridge, reprint of the 1993 original Munn [1955] Munn WD (1955) On semigroup algebras. Proc Cambridge Philos Soc 51:1–15 Munn [1957a] Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Martin P, Mazorchuk V (2014) On the representation theory of partial Brauer algebras. Q J Math 65(1):225–247. 10.1093/qmath/has043, URL https://doi.org/10.1093/qmath/has043 Martin [1990] Martin PP (1990) Representations of graph Temperley-Lieb algebras. Publ Res Inst Math Sci 26(3):485–503. 10.2977/prims/1195170958, URL https://doi.org/10.2977/prims/1195170958 Martin [2008] Martin S (2008) Schur algebras and representation theory, Cambridge Tracts in Mathematics, vol 112. Cambridge University Press, Cambridge, reprint of the 1993 original Munn [1955] Munn WD (1955) On semigroup algebras. Proc Cambridge Philos Soc 51:1–15 Munn [1957a] Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Martin PP (1990) Representations of graph Temperley-Lieb algebras. Publ Res Inst Math Sci 26(3):485–503. 10.2977/prims/1195170958, URL https://doi.org/10.2977/prims/1195170958 Martin [2008] Martin S (2008) Schur algebras and representation theory, Cambridge Tracts in Mathematics, vol 112. Cambridge University Press, Cambridge, reprint of the 1993 original Munn [1955] Munn WD (1955) On semigroup algebras. Proc Cambridge Philos Soc 51:1–15 Munn [1957a] Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Martin S (2008) Schur algebras and representation theory, Cambridge Tracts in Mathematics, vol 112. Cambridge University Press, Cambridge, reprint of the 1993 original Munn [1955] Munn WD (1955) On semigroup algebras. Proc Cambridge Philos Soc 51:1–15 Munn [1957a] Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Munn WD (1955) On semigroup algebras. Proc Cambridge Philos Soc 51:1–15 Munn [1957a] Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. 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Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Halverson T, Ram A (2005) Partition algebras. European J Combin 26(6):869–921. 10.1016/j.ejc.2004.06.005, URL http://dx.doi.org/10.1016/j.ejc.2004.06.005 Jacobson [1989] Jacobson N (1989) Basic algebra. II, 2nd edn. W. H. Freeman and Company, New York Jones [1994] Jones VFR (1994) The Potts model and the symmetric group. In: Subfactors (Kyuzeso, 1993). World Sci. Publ., River Edge, NJ, p 259–267 Kudryavtseva and Mazorchuk [2008] Kudryavtseva G, Mazorchuk V (2008) Schur-Weyl dualities for symmetric inverse semigroups. J Pure Appl Algebra 212(8):1987–1995. 10.1016/j.jpaa.2007.12.004, URL https://doi.org/10.1016/j.jpaa.2007.12.004 Lawson [1998] Lawson MV (1998) Inverse semigroups. World Scientific Publishing Co. Inc., River Edge, NJ, 10.1142/9789812816689, URL http://dx.doi.org/10.1142/9789812816689, the theory of partial symmetries Martin [1994] Martin P (1994) Temperley-Lieb algebras for nonplanar statistical mechanics—the partition algebra construction. J Knot Theory Ramifications 3(1):51–82. 10.1142/S0218216594000071, URL http://dx.doi.org/10.1142/S0218216594000071 Martin [1996] Martin P (1996) The structure of the partition algebras. J Algebra 183(2):319–358. 10.1006/jabr.1996.0223, URL http://dx.doi.org/10.1006/jabr.1996.0223 Martin and Mazorchuk [2014] Martin P, Mazorchuk V (2014) On the representation theory of partial Brauer algebras. Q J Math 65(1):225–247. 10.1093/qmath/has043, URL https://doi.org/10.1093/qmath/has043 Martin [1990] Martin PP (1990) Representations of graph Temperley-Lieb algebras. Publ Res Inst Math Sci 26(3):485–503. 10.2977/prims/1195170958, URL https://doi.org/10.2977/prims/1195170958 Martin [2008] Martin S (2008) Schur algebras and representation theory, Cambridge Tracts in Mathematics, vol 112. Cambridge University Press, Cambridge, reprint of the 1993 original Munn [1955] Munn WD (1955) On semigroup algebras. Proc Cambridge Philos Soc 51:1–15 Munn [1957a] Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Jacobson N (1989) Basic algebra. II, 2nd edn. W. H. Freeman and Company, New York Jones [1994] Jones VFR (1994) The Potts model and the symmetric group. In: Subfactors (Kyuzeso, 1993). World Sci. Publ., River Edge, NJ, p 259–267 Kudryavtseva and Mazorchuk [2008] Kudryavtseva G, Mazorchuk V (2008) Schur-Weyl dualities for symmetric inverse semigroups. J Pure Appl Algebra 212(8):1987–1995. 10.1016/j.jpaa.2007.12.004, URL https://doi.org/10.1016/j.jpaa.2007.12.004 Lawson [1998] Lawson MV (1998) Inverse semigroups. World Scientific Publishing Co. Inc., River Edge, NJ, 10.1142/9789812816689, URL http://dx.doi.org/10.1142/9789812816689, the theory of partial symmetries Martin [1994] Martin P (1994) Temperley-Lieb algebras for nonplanar statistical mechanics—the partition algebra construction. J Knot Theory Ramifications 3(1):51–82. 10.1142/S0218216594000071, URL http://dx.doi.org/10.1142/S0218216594000071 Martin [1996] Martin P (1996) The structure of the partition algebras. J Algebra 183(2):319–358. 10.1006/jabr.1996.0223, URL http://dx.doi.org/10.1006/jabr.1996.0223 Martin and Mazorchuk [2014] Martin P, Mazorchuk V (2014) On the representation theory of partial Brauer algebras. Q J Math 65(1):225–247. 10.1093/qmath/has043, URL https://doi.org/10.1093/qmath/has043 Martin [1990] Martin PP (1990) Representations of graph Temperley-Lieb algebras. Publ Res Inst Math Sci 26(3):485–503. 10.2977/prims/1195170958, URL https://doi.org/10.2977/prims/1195170958 Martin [2008] Martin S (2008) Schur algebras and representation theory, Cambridge Tracts in Mathematics, vol 112. Cambridge University Press, Cambridge, reprint of the 1993 original Munn [1955] Munn WD (1955) On semigroup algebras. Proc Cambridge Philos Soc 51:1–15 Munn [1957a] Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Jones VFR (1994) The Potts model and the symmetric group. In: Subfactors (Kyuzeso, 1993). World Sci. Publ., River Edge, NJ, p 259–267 Kudryavtseva and Mazorchuk [2008] Kudryavtseva G, Mazorchuk V (2008) Schur-Weyl dualities for symmetric inverse semigroups. J Pure Appl Algebra 212(8):1987–1995. 10.1016/j.jpaa.2007.12.004, URL https://doi.org/10.1016/j.jpaa.2007.12.004 Lawson [1998] Lawson MV (1998) Inverse semigroups. World Scientific Publishing Co. Inc., River Edge, NJ, 10.1142/9789812816689, URL http://dx.doi.org/10.1142/9789812816689, the theory of partial symmetries Martin [1994] Martin P (1994) Temperley-Lieb algebras for nonplanar statistical mechanics—the partition algebra construction. J Knot Theory Ramifications 3(1):51–82. 10.1142/S0218216594000071, URL http://dx.doi.org/10.1142/S0218216594000071 Martin [1996] Martin P (1996) The structure of the partition algebras. J Algebra 183(2):319–358. 10.1006/jabr.1996.0223, URL http://dx.doi.org/10.1006/jabr.1996.0223 Martin and Mazorchuk [2014] Martin P, Mazorchuk V (2014) On the representation theory of partial Brauer algebras. Q J Math 65(1):225–247. 10.1093/qmath/has043, URL https://doi.org/10.1093/qmath/has043 Martin [1990] Martin PP (1990) Representations of graph Temperley-Lieb algebras. Publ Res Inst Math Sci 26(3):485–503. 10.2977/prims/1195170958, URL https://doi.org/10.2977/prims/1195170958 Martin [2008] Martin S (2008) Schur algebras and representation theory, Cambridge Tracts in Mathematics, vol 112. Cambridge University Press, Cambridge, reprint of the 1993 original Munn [1955] Munn WD (1955) On semigroup algebras. Proc Cambridge Philos Soc 51:1–15 Munn [1957a] Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Kudryavtseva G, Mazorchuk V (2008) Schur-Weyl dualities for symmetric inverse semigroups. J Pure Appl Algebra 212(8):1987–1995. 10.1016/j.jpaa.2007.12.004, URL https://doi.org/10.1016/j.jpaa.2007.12.004 Lawson [1998] Lawson MV (1998) Inverse semigroups. World Scientific Publishing Co. Inc., River Edge, NJ, 10.1142/9789812816689, URL http://dx.doi.org/10.1142/9789812816689, the theory of partial symmetries Martin [1994] Martin P (1994) Temperley-Lieb algebras for nonplanar statistical mechanics—the partition algebra construction. J Knot Theory Ramifications 3(1):51–82. 10.1142/S0218216594000071, URL http://dx.doi.org/10.1142/S0218216594000071 Martin [1996] Martin P (1996) The structure of the partition algebras. J Algebra 183(2):319–358. 10.1006/jabr.1996.0223, URL http://dx.doi.org/10.1006/jabr.1996.0223 Martin and Mazorchuk [2014] Martin P, Mazorchuk V (2014) On the representation theory of partial Brauer algebras. Q J Math 65(1):225–247. 10.1093/qmath/has043, URL https://doi.org/10.1093/qmath/has043 Martin [1990] Martin PP (1990) Representations of graph Temperley-Lieb algebras. Publ Res Inst Math Sci 26(3):485–503. 10.2977/prims/1195170958, URL https://doi.org/10.2977/prims/1195170958 Martin [2008] Martin S (2008) Schur algebras and representation theory, Cambridge Tracts in Mathematics, vol 112. Cambridge University Press, Cambridge, reprint of the 1993 original Munn [1955] Munn WD (1955) On semigroup algebras. Proc Cambridge Philos Soc 51:1–15 Munn [1957a] Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Lawson MV (1998) Inverse semigroups. World Scientific Publishing Co. Inc., River Edge, NJ, 10.1142/9789812816689, URL http://dx.doi.org/10.1142/9789812816689, the theory of partial symmetries Martin [1994] Martin P (1994) Temperley-Lieb algebras for nonplanar statistical mechanics—the partition algebra construction. J Knot Theory Ramifications 3(1):51–82. 10.1142/S0218216594000071, URL http://dx.doi.org/10.1142/S0218216594000071 Martin [1996] Martin P (1996) The structure of the partition algebras. J Algebra 183(2):319–358. 10.1006/jabr.1996.0223, URL http://dx.doi.org/10.1006/jabr.1996.0223 Martin and Mazorchuk [2014] Martin P, Mazorchuk V (2014) On the representation theory of partial Brauer algebras. Q J Math 65(1):225–247. 10.1093/qmath/has043, URL https://doi.org/10.1093/qmath/has043 Martin [1990] Martin PP (1990) Representations of graph Temperley-Lieb algebras. Publ Res Inst Math Sci 26(3):485–503. 10.2977/prims/1195170958, URL https://doi.org/10.2977/prims/1195170958 Martin [2008] Martin S (2008) Schur algebras and representation theory, Cambridge Tracts in Mathematics, vol 112. Cambridge University Press, Cambridge, reprint of the 1993 original Munn [1955] Munn WD (1955) On semigroup algebras. Proc Cambridge Philos Soc 51:1–15 Munn [1957a] Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Martin P (1994) Temperley-Lieb algebras for nonplanar statistical mechanics—the partition algebra construction. J Knot Theory Ramifications 3(1):51–82. 10.1142/S0218216594000071, URL http://dx.doi.org/10.1142/S0218216594000071 Martin [1996] Martin P (1996) The structure of the partition algebras. J Algebra 183(2):319–358. 10.1006/jabr.1996.0223, URL http://dx.doi.org/10.1006/jabr.1996.0223 Martin and Mazorchuk [2014] Martin P, Mazorchuk V (2014) On the representation theory of partial Brauer algebras. Q J Math 65(1):225–247. 10.1093/qmath/has043, URL https://doi.org/10.1093/qmath/has043 Martin [1990] Martin PP (1990) Representations of graph Temperley-Lieb algebras. Publ Res Inst Math Sci 26(3):485–503. 10.2977/prims/1195170958, URL https://doi.org/10.2977/prims/1195170958 Martin [2008] Martin S (2008) Schur algebras and representation theory, Cambridge Tracts in Mathematics, vol 112. Cambridge University Press, Cambridge, reprint of the 1993 original Munn [1955] Munn WD (1955) On semigroup algebras. Proc Cambridge Philos Soc 51:1–15 Munn [1957a] Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Martin P (1996) The structure of the partition algebras. J Algebra 183(2):319–358. 10.1006/jabr.1996.0223, URL http://dx.doi.org/10.1006/jabr.1996.0223 Martin and Mazorchuk [2014] Martin P, Mazorchuk V (2014) On the representation theory of partial Brauer algebras. Q J Math 65(1):225–247. 10.1093/qmath/has043, URL https://doi.org/10.1093/qmath/has043 Martin [1990] Martin PP (1990) Representations of graph Temperley-Lieb algebras. Publ Res Inst Math Sci 26(3):485–503. 10.2977/prims/1195170958, URL https://doi.org/10.2977/prims/1195170958 Martin [2008] Martin S (2008) Schur algebras and representation theory, Cambridge Tracts in Mathematics, vol 112. Cambridge University Press, Cambridge, reprint of the 1993 original Munn [1955] Munn WD (1955) On semigroup algebras. Proc Cambridge Philos Soc 51:1–15 Munn [1957a] Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Martin P, Mazorchuk V (2014) On the representation theory of partial Brauer algebras. Q J Math 65(1):225–247. 10.1093/qmath/has043, URL https://doi.org/10.1093/qmath/has043 Martin [1990] Martin PP (1990) Representations of graph Temperley-Lieb algebras. Publ Res Inst Math Sci 26(3):485–503. 10.2977/prims/1195170958, URL https://doi.org/10.2977/prims/1195170958 Martin [2008] Martin S (2008) Schur algebras and representation theory, Cambridge Tracts in Mathematics, vol 112. Cambridge University Press, Cambridge, reprint of the 1993 original Munn [1955] Munn WD (1955) On semigroup algebras. Proc Cambridge Philos Soc 51:1–15 Munn [1957a] Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Martin PP (1990) Representations of graph Temperley-Lieb algebras. Publ Res Inst Math Sci 26(3):485–503. 10.2977/prims/1195170958, URL https://doi.org/10.2977/prims/1195170958 Martin [2008] Martin S (2008) Schur algebras and representation theory, Cambridge Tracts in Mathematics, vol 112. Cambridge University Press, Cambridge, reprint of the 1993 original Munn [1955] Munn WD (1955) On semigroup algebras. Proc Cambridge Philos Soc 51:1–15 Munn [1957a] Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Martin S (2008) Schur algebras and representation theory, Cambridge Tracts in Mathematics, vol 112. Cambridge University Press, Cambridge, reprint of the 1993 original Munn [1955] Munn WD (1955) On semigroup algebras. Proc Cambridge Philos Soc 51:1–15 Munn [1957a] Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Munn WD (1955) On semigroup algebras. Proc Cambridge Philos Soc 51:1–15 Munn [1957a] Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J.
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J Knot Theory Ramifications 3(1):51–82. 10.1142/S0218216594000071, URL http://dx.doi.org/10.1142/S0218216594000071 Martin [1996] Martin P (1996) The structure of the partition algebras. J Algebra 183(2):319–358. 10.1006/jabr.1996.0223, URL http://dx.doi.org/10.1006/jabr.1996.0223 Martin and Mazorchuk [2014] Martin P, Mazorchuk V (2014) On the representation theory of partial Brauer algebras. Q J Math 65(1):225–247. 10.1093/qmath/has043, URL https://doi.org/10.1093/qmath/has043 Martin [1990] Martin PP (1990) Representations of graph Temperley-Lieb algebras. Publ Res Inst Math Sci 26(3):485–503. 10.2977/prims/1195170958, URL https://doi.org/10.2977/prims/1195170958 Martin [2008] Martin S (2008) Schur algebras and representation theory, Cambridge Tracts in Mathematics, vol 112. Cambridge University Press, Cambridge, reprint of the 1993 original Munn [1955] Munn WD (1955) On semigroup algebras. Proc Cambridge Philos Soc 51:1–15 Munn [1957a] Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. 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J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Jacobson N (1989) Basic algebra. II, 2nd edn. W. H. Freeman and Company, New York Jones [1994] Jones VFR (1994) The Potts model and the symmetric group. In: Subfactors (Kyuzeso, 1993). World Sci. Publ., River Edge, NJ, p 259–267 Kudryavtseva and Mazorchuk [2008] Kudryavtseva G, Mazorchuk V (2008) Schur-Weyl dualities for symmetric inverse semigroups. J Pure Appl Algebra 212(8):1987–1995. 10.1016/j.jpaa.2007.12.004, URL https://doi.org/10.1016/j.jpaa.2007.12.004 Lawson [1998] Lawson MV (1998) Inverse semigroups. World Scientific Publishing Co. Inc., River Edge, NJ, 10.1142/9789812816689, URL http://dx.doi.org/10.1142/9789812816689, the theory of partial symmetries Martin [1994] Martin P (1994) Temperley-Lieb algebras for nonplanar statistical mechanics—the partition algebra construction. J Knot Theory Ramifications 3(1):51–82. 10.1142/S0218216594000071, URL http://dx.doi.org/10.1142/S0218216594000071 Martin [1996] Martin P (1996) The structure of the partition algebras. J Algebra 183(2):319–358. 10.1006/jabr.1996.0223, URL http://dx.doi.org/10.1006/jabr.1996.0223 Martin and Mazorchuk [2014] Martin P, Mazorchuk V (2014) On the representation theory of partial Brauer algebras. Q J Math 65(1):225–247. 10.1093/qmath/has043, URL https://doi.org/10.1093/qmath/has043 Martin [1990] Martin PP (1990) Representations of graph Temperley-Lieb algebras. Publ Res Inst Math Sci 26(3):485–503. 10.2977/prims/1195170958, URL https://doi.org/10.2977/prims/1195170958 Martin [2008] Martin S (2008) Schur algebras and representation theory, Cambridge Tracts in Mathematics, vol 112. Cambridge University Press, Cambridge, reprint of the 1993 original Munn [1955] Munn WD (1955) On semigroup algebras. Proc Cambridge Philos Soc 51:1–15 Munn [1957a] Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Jones VFR (1994) The Potts model and the symmetric group. In: Subfactors (Kyuzeso, 1993). World Sci. Publ., River Edge, NJ, p 259–267 Kudryavtseva and Mazorchuk [2008] Kudryavtseva G, Mazorchuk V (2008) Schur-Weyl dualities for symmetric inverse semigroups. J Pure Appl Algebra 212(8):1987–1995. 10.1016/j.jpaa.2007.12.004, URL https://doi.org/10.1016/j.jpaa.2007.12.004 Lawson [1998] Lawson MV (1998) Inverse semigroups. World Scientific Publishing Co. Inc., River Edge, NJ, 10.1142/9789812816689, URL http://dx.doi.org/10.1142/9789812816689, the theory of partial symmetries Martin [1994] Martin P (1994) Temperley-Lieb algebras for nonplanar statistical mechanics—the partition algebra construction. J Knot Theory Ramifications 3(1):51–82. 10.1142/S0218216594000071, URL http://dx.doi.org/10.1142/S0218216594000071 Martin [1996] Martin P (1996) The structure of the partition algebras. J Algebra 183(2):319–358. 10.1006/jabr.1996.0223, URL http://dx.doi.org/10.1006/jabr.1996.0223 Martin and Mazorchuk [2014] Martin P, Mazorchuk V (2014) On the representation theory of partial Brauer algebras. Q J Math 65(1):225–247. 10.1093/qmath/has043, URL https://doi.org/10.1093/qmath/has043 Martin [1990] Martin PP (1990) Representations of graph Temperley-Lieb algebras. Publ Res Inst Math Sci 26(3):485–503. 10.2977/prims/1195170958, URL https://doi.org/10.2977/prims/1195170958 Martin [2008] Martin S (2008) Schur algebras and representation theory, Cambridge Tracts in Mathematics, vol 112. Cambridge University Press, Cambridge, reprint of the 1993 original Munn [1955] Munn WD (1955) On semigroup algebras. Proc Cambridge Philos Soc 51:1–15 Munn [1957a] Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Kudryavtseva G, Mazorchuk V (2008) Schur-Weyl dualities for symmetric inverse semigroups. J Pure Appl Algebra 212(8):1987–1995. 10.1016/j.jpaa.2007.12.004, URL https://doi.org/10.1016/j.jpaa.2007.12.004 Lawson [1998] Lawson MV (1998) Inverse semigroups. World Scientific Publishing Co. Inc., River Edge, NJ, 10.1142/9789812816689, URL http://dx.doi.org/10.1142/9789812816689, the theory of partial symmetries Martin [1994] Martin P (1994) Temperley-Lieb algebras for nonplanar statistical mechanics—the partition algebra construction. J Knot Theory Ramifications 3(1):51–82. 10.1142/S0218216594000071, URL http://dx.doi.org/10.1142/S0218216594000071 Martin [1996] Martin P (1996) The structure of the partition algebras. J Algebra 183(2):319–358. 10.1006/jabr.1996.0223, URL http://dx.doi.org/10.1006/jabr.1996.0223 Martin and Mazorchuk [2014] Martin P, Mazorchuk V (2014) On the representation theory of partial Brauer algebras. Q J Math 65(1):225–247. 10.1093/qmath/has043, URL https://doi.org/10.1093/qmath/has043 Martin [1990] Martin PP (1990) Representations of graph Temperley-Lieb algebras. Publ Res Inst Math Sci 26(3):485–503. 10.2977/prims/1195170958, URL https://doi.org/10.2977/prims/1195170958 Martin [2008] Martin S (2008) Schur algebras and representation theory, Cambridge Tracts in Mathematics, vol 112. Cambridge University Press, Cambridge, reprint of the 1993 original Munn [1955] Munn WD (1955) On semigroup algebras. Proc Cambridge Philos Soc 51:1–15 Munn [1957a] Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Lawson MV (1998) Inverse semigroups. World Scientific Publishing Co. Inc., River Edge, NJ, 10.1142/9789812816689, URL http://dx.doi.org/10.1142/9789812816689, the theory of partial symmetries Martin [1994] Martin P (1994) Temperley-Lieb algebras for nonplanar statistical mechanics—the partition algebra construction. J Knot Theory Ramifications 3(1):51–82. 10.1142/S0218216594000071, URL http://dx.doi.org/10.1142/S0218216594000071 Martin [1996] Martin P (1996) The structure of the partition algebras. J Algebra 183(2):319–358. 10.1006/jabr.1996.0223, URL http://dx.doi.org/10.1006/jabr.1996.0223 Martin and Mazorchuk [2014] Martin P, Mazorchuk V (2014) On the representation theory of partial Brauer algebras. Q J Math 65(1):225–247. 10.1093/qmath/has043, URL https://doi.org/10.1093/qmath/has043 Martin [1990] Martin PP (1990) Representations of graph Temperley-Lieb algebras. Publ Res Inst Math Sci 26(3):485–503. 10.2977/prims/1195170958, URL https://doi.org/10.2977/prims/1195170958 Martin [2008] Martin S (2008) Schur algebras and representation theory, Cambridge Tracts in Mathematics, vol 112. Cambridge University Press, Cambridge, reprint of the 1993 original Munn [1955] Munn WD (1955) On semigroup algebras. Proc Cambridge Philos Soc 51:1–15 Munn [1957a] Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Martin P (1994) Temperley-Lieb algebras for nonplanar statistical mechanics—the partition algebra construction. J Knot Theory Ramifications 3(1):51–82. 10.1142/S0218216594000071, URL http://dx.doi.org/10.1142/S0218216594000071 Martin [1996] Martin P (1996) The structure of the partition algebras. J Algebra 183(2):319–358. 10.1006/jabr.1996.0223, URL http://dx.doi.org/10.1006/jabr.1996.0223 Martin and Mazorchuk [2014] Martin P, Mazorchuk V (2014) On the representation theory of partial Brauer algebras. Q J Math 65(1):225–247. 10.1093/qmath/has043, URL https://doi.org/10.1093/qmath/has043 Martin [1990] Martin PP (1990) Representations of graph Temperley-Lieb algebras. Publ Res Inst Math Sci 26(3):485–503. 10.2977/prims/1195170958, URL https://doi.org/10.2977/prims/1195170958 Martin [2008] Martin S (2008) Schur algebras and representation theory, Cambridge Tracts in Mathematics, vol 112. Cambridge University Press, Cambridge, reprint of the 1993 original Munn [1955] Munn WD (1955) On semigroup algebras. Proc Cambridge Philos Soc 51:1–15 Munn [1957a] Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Martin P (1996) The structure of the partition algebras. J Algebra 183(2):319–358. 10.1006/jabr.1996.0223, URL http://dx.doi.org/10.1006/jabr.1996.0223 Martin and Mazorchuk [2014] Martin P, Mazorchuk V (2014) On the representation theory of partial Brauer algebras. Q J Math 65(1):225–247. 10.1093/qmath/has043, URL https://doi.org/10.1093/qmath/has043 Martin [1990] Martin PP (1990) Representations of graph Temperley-Lieb algebras. Publ Res Inst Math Sci 26(3):485–503. 10.2977/prims/1195170958, URL https://doi.org/10.2977/prims/1195170958 Martin [2008] Martin S (2008) Schur algebras and representation theory, Cambridge Tracts in Mathematics, vol 112. Cambridge University Press, Cambridge, reprint of the 1993 original Munn [1955] Munn WD (1955) On semigroup algebras. Proc Cambridge Philos Soc 51:1–15 Munn [1957a] Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Martin P, Mazorchuk V (2014) On the representation theory of partial Brauer algebras. Q J Math 65(1):225–247. 10.1093/qmath/has043, URL https://doi.org/10.1093/qmath/has043 Martin [1990] Martin PP (1990) Representations of graph Temperley-Lieb algebras. Publ Res Inst Math Sci 26(3):485–503. 10.2977/prims/1195170958, URL https://doi.org/10.2977/prims/1195170958 Martin [2008] Martin S (2008) Schur algebras and representation theory, Cambridge Tracts in Mathematics, vol 112. Cambridge University Press, Cambridge, reprint of the 1993 original Munn [1955] Munn WD (1955) On semigroup algebras. Proc Cambridge Philos Soc 51:1–15 Munn [1957a] Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Martin PP (1990) Representations of graph Temperley-Lieb algebras. Publ Res Inst Math Sci 26(3):485–503. 10.2977/prims/1195170958, URL https://doi.org/10.2977/prims/1195170958 Martin [2008] Martin S (2008) Schur algebras and representation theory, Cambridge Tracts in Mathematics, vol 112. Cambridge University Press, Cambridge, reprint of the 1993 original Munn [1955] Munn WD (1955) On semigroup algebras. Proc Cambridge Philos Soc 51:1–15 Munn [1957a] Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Martin S (2008) Schur algebras and representation theory, Cambridge Tracts in Mathematics, vol 112. Cambridge University Press, Cambridge, reprint of the 1993 original Munn [1955] Munn WD (1955) On semigroup algebras. Proc Cambridge Philos Soc 51:1–15 Munn [1957a] Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Munn WD (1955) On semigroup algebras. Proc Cambridge Philos Soc 51:1–15 Munn [1957a] Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J.
- Jacobson N (1989) Basic algebra. II, 2nd edn. W. H. Freeman and Company, New York Jones [1994] Jones VFR (1994) The Potts model and the symmetric group. In: Subfactors (Kyuzeso, 1993). World Sci. Publ., River Edge, NJ, p 259–267 Kudryavtseva and Mazorchuk [2008] Kudryavtseva G, Mazorchuk V (2008) Schur-Weyl dualities for symmetric inverse semigroups. J Pure Appl Algebra 212(8):1987–1995. 10.1016/j.jpaa.2007.12.004, URL https://doi.org/10.1016/j.jpaa.2007.12.004 Lawson [1998] Lawson MV (1998) Inverse semigroups. World Scientific Publishing Co. Inc., River Edge, NJ, 10.1142/9789812816689, URL http://dx.doi.org/10.1142/9789812816689, the theory of partial symmetries Martin [1994] Martin P (1994) Temperley-Lieb algebras for nonplanar statistical mechanics—the partition algebra construction. J Knot Theory Ramifications 3(1):51–82. 10.1142/S0218216594000071, URL http://dx.doi.org/10.1142/S0218216594000071 Martin [1996] Martin P (1996) The structure of the partition algebras. J Algebra 183(2):319–358. 10.1006/jabr.1996.0223, URL http://dx.doi.org/10.1006/jabr.1996.0223 Martin and Mazorchuk [2014] Martin P, Mazorchuk V (2014) On the representation theory of partial Brauer algebras. Q J Math 65(1):225–247. 10.1093/qmath/has043, URL https://doi.org/10.1093/qmath/has043 Martin [1990] Martin PP (1990) Representations of graph Temperley-Lieb algebras. Publ Res Inst Math Sci 26(3):485–503. 10.2977/prims/1195170958, URL https://doi.org/10.2977/prims/1195170958 Martin [2008] Martin S (2008) Schur algebras and representation theory, Cambridge Tracts in Mathematics, vol 112. Cambridge University Press, Cambridge, reprint of the 1993 original Munn [1955] Munn WD (1955) On semigroup algebras. Proc Cambridge Philos Soc 51:1–15 Munn [1957a] Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Jones VFR (1994) The Potts model and the symmetric group. In: Subfactors (Kyuzeso, 1993). World Sci. Publ., River Edge, NJ, p 259–267 Kudryavtseva and Mazorchuk [2008] Kudryavtseva G, Mazorchuk V (2008) Schur-Weyl dualities for symmetric inverse semigroups. J Pure Appl Algebra 212(8):1987–1995. 10.1016/j.jpaa.2007.12.004, URL https://doi.org/10.1016/j.jpaa.2007.12.004 Lawson [1998] Lawson MV (1998) Inverse semigroups. World Scientific Publishing Co. Inc., River Edge, NJ, 10.1142/9789812816689, URL http://dx.doi.org/10.1142/9789812816689, the theory of partial symmetries Martin [1994] Martin P (1994) Temperley-Lieb algebras for nonplanar statistical mechanics—the partition algebra construction. J Knot Theory Ramifications 3(1):51–82. 10.1142/S0218216594000071, URL http://dx.doi.org/10.1142/S0218216594000071 Martin [1996] Martin P (1996) The structure of the partition algebras. J Algebra 183(2):319–358. 10.1006/jabr.1996.0223, URL http://dx.doi.org/10.1006/jabr.1996.0223 Martin and Mazorchuk [2014] Martin P, Mazorchuk V (2014) On the representation theory of partial Brauer algebras. Q J Math 65(1):225–247. 10.1093/qmath/has043, URL https://doi.org/10.1093/qmath/has043 Martin [1990] Martin PP (1990) Representations of graph Temperley-Lieb algebras. Publ Res Inst Math Sci 26(3):485–503. 10.2977/prims/1195170958, URL https://doi.org/10.2977/prims/1195170958 Martin [2008] Martin S (2008) Schur algebras and representation theory, Cambridge Tracts in Mathematics, vol 112. Cambridge University Press, Cambridge, reprint of the 1993 original Munn [1955] Munn WD (1955) On semigroup algebras. Proc Cambridge Philos Soc 51:1–15 Munn [1957a] Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Kudryavtseva G, Mazorchuk V (2008) Schur-Weyl dualities for symmetric inverse semigroups. J Pure Appl Algebra 212(8):1987–1995. 10.1016/j.jpaa.2007.12.004, URL https://doi.org/10.1016/j.jpaa.2007.12.004 Lawson [1998] Lawson MV (1998) Inverse semigroups. World Scientific Publishing Co. Inc., River Edge, NJ, 10.1142/9789812816689, URL http://dx.doi.org/10.1142/9789812816689, the theory of partial symmetries Martin [1994] Martin P (1994) Temperley-Lieb algebras for nonplanar statistical mechanics—the partition algebra construction. J Knot Theory Ramifications 3(1):51–82. 10.1142/S0218216594000071, URL http://dx.doi.org/10.1142/S0218216594000071 Martin [1996] Martin P (1996) The structure of the partition algebras. J Algebra 183(2):319–358. 10.1006/jabr.1996.0223, URL http://dx.doi.org/10.1006/jabr.1996.0223 Martin and Mazorchuk [2014] Martin P, Mazorchuk V (2014) On the representation theory of partial Brauer algebras. Q J Math 65(1):225–247. 10.1093/qmath/has043, URL https://doi.org/10.1093/qmath/has043 Martin [1990] Martin PP (1990) Representations of graph Temperley-Lieb algebras. Publ Res Inst Math Sci 26(3):485–503. 10.2977/prims/1195170958, URL https://doi.org/10.2977/prims/1195170958 Martin [2008] Martin S (2008) Schur algebras and representation theory, Cambridge Tracts in Mathematics, vol 112. Cambridge University Press, Cambridge, reprint of the 1993 original Munn [1955] Munn WD (1955) On semigroup algebras. Proc Cambridge Philos Soc 51:1–15 Munn [1957a] Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Lawson MV (1998) Inverse semigroups. World Scientific Publishing Co. Inc., River Edge, NJ, 10.1142/9789812816689, URL http://dx.doi.org/10.1142/9789812816689, the theory of partial symmetries Martin [1994] Martin P (1994) Temperley-Lieb algebras for nonplanar statistical mechanics—the partition algebra construction. J Knot Theory Ramifications 3(1):51–82. 10.1142/S0218216594000071, URL http://dx.doi.org/10.1142/S0218216594000071 Martin [1996] Martin P (1996) The structure of the partition algebras. J Algebra 183(2):319–358. 10.1006/jabr.1996.0223, URL http://dx.doi.org/10.1006/jabr.1996.0223 Martin and Mazorchuk [2014] Martin P, Mazorchuk V (2014) On the representation theory of partial Brauer algebras. Q J Math 65(1):225–247. 10.1093/qmath/has043, URL https://doi.org/10.1093/qmath/has043 Martin [1990] Martin PP (1990) Representations of graph Temperley-Lieb algebras. Publ Res Inst Math Sci 26(3):485–503. 10.2977/prims/1195170958, URL https://doi.org/10.2977/prims/1195170958 Martin [2008] Martin S (2008) Schur algebras and representation theory, Cambridge Tracts in Mathematics, vol 112. Cambridge University Press, Cambridge, reprint of the 1993 original Munn [1955] Munn WD (1955) On semigroup algebras. Proc Cambridge Philos Soc 51:1–15 Munn [1957a] Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Martin P (1994) Temperley-Lieb algebras for nonplanar statistical mechanics—the partition algebra construction. J Knot Theory Ramifications 3(1):51–82. 10.1142/S0218216594000071, URL http://dx.doi.org/10.1142/S0218216594000071 Martin [1996] Martin P (1996) The structure of the partition algebras. J Algebra 183(2):319–358. 10.1006/jabr.1996.0223, URL http://dx.doi.org/10.1006/jabr.1996.0223 Martin and Mazorchuk [2014] Martin P, Mazorchuk V (2014) On the representation theory of partial Brauer algebras. Q J Math 65(1):225–247. 10.1093/qmath/has043, URL https://doi.org/10.1093/qmath/has043 Martin [1990] Martin PP (1990) Representations of graph Temperley-Lieb algebras. Publ Res Inst Math Sci 26(3):485–503. 10.2977/prims/1195170958, URL https://doi.org/10.2977/prims/1195170958 Martin [2008] Martin S (2008) Schur algebras and representation theory, Cambridge Tracts in Mathematics, vol 112. Cambridge University Press, Cambridge, reprint of the 1993 original Munn [1955] Munn WD (1955) On semigroup algebras. Proc Cambridge Philos Soc 51:1–15 Munn [1957a] Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Martin P (1996) The structure of the partition algebras. J Algebra 183(2):319–358. 10.1006/jabr.1996.0223, URL http://dx.doi.org/10.1006/jabr.1996.0223 Martin and Mazorchuk [2014] Martin P, Mazorchuk V (2014) On the representation theory of partial Brauer algebras. Q J Math 65(1):225–247. 10.1093/qmath/has043, URL https://doi.org/10.1093/qmath/has043 Martin [1990] Martin PP (1990) Representations of graph Temperley-Lieb algebras. Publ Res Inst Math Sci 26(3):485–503. 10.2977/prims/1195170958, URL https://doi.org/10.2977/prims/1195170958 Martin [2008] Martin S (2008) Schur algebras and representation theory, Cambridge Tracts in Mathematics, vol 112. Cambridge University Press, Cambridge, reprint of the 1993 original Munn [1955] Munn WD (1955) On semigroup algebras. Proc Cambridge Philos Soc 51:1–15 Munn [1957a] Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Martin P, Mazorchuk V (2014) On the representation theory of partial Brauer algebras. Q J Math 65(1):225–247. 10.1093/qmath/has043, URL https://doi.org/10.1093/qmath/has043 Martin [1990] Martin PP (1990) Representations of graph Temperley-Lieb algebras. Publ Res Inst Math Sci 26(3):485–503. 10.2977/prims/1195170958, URL https://doi.org/10.2977/prims/1195170958 Martin [2008] Martin S (2008) Schur algebras and representation theory, Cambridge Tracts in Mathematics, vol 112. Cambridge University Press, Cambridge, reprint of the 1993 original Munn [1955] Munn WD (1955) On semigroup algebras. Proc Cambridge Philos Soc 51:1–15 Munn [1957a] Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Martin PP (1990) Representations of graph Temperley-Lieb algebras. Publ Res Inst Math Sci 26(3):485–503. 10.2977/prims/1195170958, URL https://doi.org/10.2977/prims/1195170958 Martin [2008] Martin S (2008) Schur algebras and representation theory, Cambridge Tracts in Mathematics, vol 112. Cambridge University Press, Cambridge, reprint of the 1993 original Munn [1955] Munn WD (1955) On semigroup algebras. Proc Cambridge Philos Soc 51:1–15 Munn [1957a] Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Martin S (2008) Schur algebras and representation theory, Cambridge Tracts in Mathematics, vol 112. Cambridge University Press, Cambridge, reprint of the 1993 original Munn [1955] Munn WD (1955) On semigroup algebras. Proc Cambridge Philos Soc 51:1–15 Munn [1957a] Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Munn WD (1955) On semigroup algebras. Proc Cambridge Philos Soc 51:1–15 Munn [1957a] Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J.
- Jones VFR (1994) The Potts model and the symmetric group. In: Subfactors (Kyuzeso, 1993). World Sci. Publ., River Edge, NJ, p 259–267 Kudryavtseva and Mazorchuk [2008] Kudryavtseva G, Mazorchuk V (2008) Schur-Weyl dualities for symmetric inverse semigroups. J Pure Appl Algebra 212(8):1987–1995. 10.1016/j.jpaa.2007.12.004, URL https://doi.org/10.1016/j.jpaa.2007.12.004 Lawson [1998] Lawson MV (1998) Inverse semigroups. World Scientific Publishing Co. Inc., River Edge, NJ, 10.1142/9789812816689, URL http://dx.doi.org/10.1142/9789812816689, the theory of partial symmetries Martin [1994] Martin P (1994) Temperley-Lieb algebras for nonplanar statistical mechanics—the partition algebra construction. J Knot Theory Ramifications 3(1):51–82. 10.1142/S0218216594000071, URL http://dx.doi.org/10.1142/S0218216594000071 Martin [1996] Martin P (1996) The structure of the partition algebras. J Algebra 183(2):319–358. 10.1006/jabr.1996.0223, URL http://dx.doi.org/10.1006/jabr.1996.0223 Martin and Mazorchuk [2014] Martin P, Mazorchuk V (2014) On the representation theory of partial Brauer algebras. Q J Math 65(1):225–247. 10.1093/qmath/has043, URL https://doi.org/10.1093/qmath/has043 Martin [1990] Martin PP (1990) Representations of graph Temperley-Lieb algebras. Publ Res Inst Math Sci 26(3):485–503. 10.2977/prims/1195170958, URL https://doi.org/10.2977/prims/1195170958 Martin [2008] Martin S (2008) Schur algebras and representation theory, Cambridge Tracts in Mathematics, vol 112. Cambridge University Press, Cambridge, reprint of the 1993 original Munn [1955] Munn WD (1955) On semigroup algebras. Proc Cambridge Philos Soc 51:1–15 Munn [1957a] Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Kudryavtseva G, Mazorchuk V (2008) Schur-Weyl dualities for symmetric inverse semigroups. J Pure Appl Algebra 212(8):1987–1995. 10.1016/j.jpaa.2007.12.004, URL https://doi.org/10.1016/j.jpaa.2007.12.004 Lawson [1998] Lawson MV (1998) Inverse semigroups. World Scientific Publishing Co. Inc., River Edge, NJ, 10.1142/9789812816689, URL http://dx.doi.org/10.1142/9789812816689, the theory of partial symmetries Martin [1994] Martin P (1994) Temperley-Lieb algebras for nonplanar statistical mechanics—the partition algebra construction. J Knot Theory Ramifications 3(1):51–82. 10.1142/S0218216594000071, URL http://dx.doi.org/10.1142/S0218216594000071 Martin [1996] Martin P (1996) The structure of the partition algebras. J Algebra 183(2):319–358. 10.1006/jabr.1996.0223, URL http://dx.doi.org/10.1006/jabr.1996.0223 Martin and Mazorchuk [2014] Martin P, Mazorchuk V (2014) On the representation theory of partial Brauer algebras. Q J Math 65(1):225–247. 10.1093/qmath/has043, URL https://doi.org/10.1093/qmath/has043 Martin [1990] Martin PP (1990) Representations of graph Temperley-Lieb algebras. Publ Res Inst Math Sci 26(3):485–503. 10.2977/prims/1195170958, URL https://doi.org/10.2977/prims/1195170958 Martin [2008] Martin S (2008) Schur algebras and representation theory, Cambridge Tracts in Mathematics, vol 112. Cambridge University Press, Cambridge, reprint of the 1993 original Munn [1955] Munn WD (1955) On semigroup algebras. Proc Cambridge Philos Soc 51:1–15 Munn [1957a] Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Lawson MV (1998) Inverse semigroups. World Scientific Publishing Co. Inc., River Edge, NJ, 10.1142/9789812816689, URL http://dx.doi.org/10.1142/9789812816689, the theory of partial symmetries Martin [1994] Martin P (1994) Temperley-Lieb algebras for nonplanar statistical mechanics—the partition algebra construction. J Knot Theory Ramifications 3(1):51–82. 10.1142/S0218216594000071, URL http://dx.doi.org/10.1142/S0218216594000071 Martin [1996] Martin P (1996) The structure of the partition algebras. J Algebra 183(2):319–358. 10.1006/jabr.1996.0223, URL http://dx.doi.org/10.1006/jabr.1996.0223 Martin and Mazorchuk [2014] Martin P, Mazorchuk V (2014) On the representation theory of partial Brauer algebras. Q J Math 65(1):225–247. 10.1093/qmath/has043, URL https://doi.org/10.1093/qmath/has043 Martin [1990] Martin PP (1990) Representations of graph Temperley-Lieb algebras. Publ Res Inst Math Sci 26(3):485–503. 10.2977/prims/1195170958, URL https://doi.org/10.2977/prims/1195170958 Martin [2008] Martin S (2008) Schur algebras and representation theory, Cambridge Tracts in Mathematics, vol 112. Cambridge University Press, Cambridge, reprint of the 1993 original Munn [1955] Munn WD (1955) On semigroup algebras. Proc Cambridge Philos Soc 51:1–15 Munn [1957a] Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Martin P (1994) Temperley-Lieb algebras for nonplanar statistical mechanics—the partition algebra construction. J Knot Theory Ramifications 3(1):51–82. 10.1142/S0218216594000071, URL http://dx.doi.org/10.1142/S0218216594000071 Martin [1996] Martin P (1996) The structure of the partition algebras. J Algebra 183(2):319–358. 10.1006/jabr.1996.0223, URL http://dx.doi.org/10.1006/jabr.1996.0223 Martin and Mazorchuk [2014] Martin P, Mazorchuk V (2014) On the representation theory of partial Brauer algebras. Q J Math 65(1):225–247. 10.1093/qmath/has043, URL https://doi.org/10.1093/qmath/has043 Martin [1990] Martin PP (1990) Representations of graph Temperley-Lieb algebras. Publ Res Inst Math Sci 26(3):485–503. 10.2977/prims/1195170958, URL https://doi.org/10.2977/prims/1195170958 Martin [2008] Martin S (2008) Schur algebras and representation theory, Cambridge Tracts in Mathematics, vol 112. Cambridge University Press, Cambridge, reprint of the 1993 original Munn [1955] Munn WD (1955) On semigroup algebras. Proc Cambridge Philos Soc 51:1–15 Munn [1957a] Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Martin P (1996) The structure of the partition algebras. J Algebra 183(2):319–358. 10.1006/jabr.1996.0223, URL http://dx.doi.org/10.1006/jabr.1996.0223 Martin and Mazorchuk [2014] Martin P, Mazorchuk V (2014) On the representation theory of partial Brauer algebras. Q J Math 65(1):225–247. 10.1093/qmath/has043, URL https://doi.org/10.1093/qmath/has043 Martin [1990] Martin PP (1990) Representations of graph Temperley-Lieb algebras. Publ Res Inst Math Sci 26(3):485–503. 10.2977/prims/1195170958, URL https://doi.org/10.2977/prims/1195170958 Martin [2008] Martin S (2008) Schur algebras and representation theory, Cambridge Tracts in Mathematics, vol 112. Cambridge University Press, Cambridge, reprint of the 1993 original Munn [1955] Munn WD (1955) On semigroup algebras. Proc Cambridge Philos Soc 51:1–15 Munn [1957a] Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Martin P, Mazorchuk V (2014) On the representation theory of partial Brauer algebras. Q J Math 65(1):225–247. 10.1093/qmath/has043, URL https://doi.org/10.1093/qmath/has043 Martin [1990] Martin PP (1990) Representations of graph Temperley-Lieb algebras. Publ Res Inst Math Sci 26(3):485–503. 10.2977/prims/1195170958, URL https://doi.org/10.2977/prims/1195170958 Martin [2008] Martin S (2008) Schur algebras and representation theory, Cambridge Tracts in Mathematics, vol 112. Cambridge University Press, Cambridge, reprint of the 1993 original Munn [1955] Munn WD (1955) On semigroup algebras. Proc Cambridge Philos Soc 51:1–15 Munn [1957a] Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Martin PP (1990) Representations of graph Temperley-Lieb algebras. Publ Res Inst Math Sci 26(3):485–503. 10.2977/prims/1195170958, URL https://doi.org/10.2977/prims/1195170958 Martin [2008] Martin S (2008) Schur algebras and representation theory, Cambridge Tracts in Mathematics, vol 112. Cambridge University Press, Cambridge, reprint of the 1993 original Munn [1955] Munn WD (1955) On semigroup algebras. Proc Cambridge Philos Soc 51:1–15 Munn [1957a] Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Martin S (2008) Schur algebras and representation theory, Cambridge Tracts in Mathematics, vol 112. Cambridge University Press, Cambridge, reprint of the 1993 original Munn [1955] Munn WD (1955) On semigroup algebras. Proc Cambridge Philos Soc 51:1–15 Munn [1957a] Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Munn WD (1955) On semigroup algebras. Proc Cambridge Philos Soc 51:1–15 Munn [1957a] Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J.
- Kudryavtseva G, Mazorchuk V (2008) Schur-Weyl dualities for symmetric inverse semigroups. J Pure Appl Algebra 212(8):1987–1995. 10.1016/j.jpaa.2007.12.004, URL https://doi.org/10.1016/j.jpaa.2007.12.004 Lawson [1998] Lawson MV (1998) Inverse semigroups. World Scientific Publishing Co. Inc., River Edge, NJ, 10.1142/9789812816689, URL http://dx.doi.org/10.1142/9789812816689, the theory of partial symmetries Martin [1994] Martin P (1994) Temperley-Lieb algebras for nonplanar statistical mechanics—the partition algebra construction. J Knot Theory Ramifications 3(1):51–82. 10.1142/S0218216594000071, URL http://dx.doi.org/10.1142/S0218216594000071 Martin [1996] Martin P (1996) The structure of the partition algebras. J Algebra 183(2):319–358. 10.1006/jabr.1996.0223, URL http://dx.doi.org/10.1006/jabr.1996.0223 Martin and Mazorchuk [2014] Martin P, Mazorchuk V (2014) On the representation theory of partial Brauer algebras. Q J Math 65(1):225–247. 10.1093/qmath/has043, URL https://doi.org/10.1093/qmath/has043 Martin [1990] Martin PP (1990) Representations of graph Temperley-Lieb algebras. Publ Res Inst Math Sci 26(3):485–503. 10.2977/prims/1195170958, URL https://doi.org/10.2977/prims/1195170958 Martin [2008] Martin S (2008) Schur algebras and representation theory, Cambridge Tracts in Mathematics, vol 112. Cambridge University Press, Cambridge, reprint of the 1993 original Munn [1955] Munn WD (1955) On semigroup algebras. Proc Cambridge Philos Soc 51:1–15 Munn [1957a] Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Lawson MV (1998) Inverse semigroups. World Scientific Publishing Co. Inc., River Edge, NJ, 10.1142/9789812816689, URL http://dx.doi.org/10.1142/9789812816689, the theory of partial symmetries Martin [1994] Martin P (1994) Temperley-Lieb algebras for nonplanar statistical mechanics—the partition algebra construction. J Knot Theory Ramifications 3(1):51–82. 10.1142/S0218216594000071, URL http://dx.doi.org/10.1142/S0218216594000071 Martin [1996] Martin P (1996) The structure of the partition algebras. J Algebra 183(2):319–358. 10.1006/jabr.1996.0223, URL http://dx.doi.org/10.1006/jabr.1996.0223 Martin and Mazorchuk [2014] Martin P, Mazorchuk V (2014) On the representation theory of partial Brauer algebras. Q J Math 65(1):225–247. 10.1093/qmath/has043, URL https://doi.org/10.1093/qmath/has043 Martin [1990] Martin PP (1990) Representations of graph Temperley-Lieb algebras. Publ Res Inst Math Sci 26(3):485–503. 10.2977/prims/1195170958, URL https://doi.org/10.2977/prims/1195170958 Martin [2008] Martin S (2008) Schur algebras and representation theory, Cambridge Tracts in Mathematics, vol 112. Cambridge University Press, Cambridge, reprint of the 1993 original Munn [1955] Munn WD (1955) On semigroup algebras. Proc Cambridge Philos Soc 51:1–15 Munn [1957a] Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Martin P (1994) Temperley-Lieb algebras for nonplanar statistical mechanics—the partition algebra construction. J Knot Theory Ramifications 3(1):51–82. 10.1142/S0218216594000071, URL http://dx.doi.org/10.1142/S0218216594000071 Martin [1996] Martin P (1996) The structure of the partition algebras. J Algebra 183(2):319–358. 10.1006/jabr.1996.0223, URL http://dx.doi.org/10.1006/jabr.1996.0223 Martin and Mazorchuk [2014] Martin P, Mazorchuk V (2014) On the representation theory of partial Brauer algebras. Q J Math 65(1):225–247. 10.1093/qmath/has043, URL https://doi.org/10.1093/qmath/has043 Martin [1990] Martin PP (1990) Representations of graph Temperley-Lieb algebras. Publ Res Inst Math Sci 26(3):485–503. 10.2977/prims/1195170958, URL https://doi.org/10.2977/prims/1195170958 Martin [2008] Martin S (2008) Schur algebras and representation theory, Cambridge Tracts in Mathematics, vol 112. Cambridge University Press, Cambridge, reprint of the 1993 original Munn [1955] Munn WD (1955) On semigroup algebras. Proc Cambridge Philos Soc 51:1–15 Munn [1957a] Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Martin P (1996) The structure of the partition algebras. J Algebra 183(2):319–358. 10.1006/jabr.1996.0223, URL http://dx.doi.org/10.1006/jabr.1996.0223 Martin and Mazorchuk [2014] Martin P, Mazorchuk V (2014) On the representation theory of partial Brauer algebras. Q J Math 65(1):225–247. 10.1093/qmath/has043, URL https://doi.org/10.1093/qmath/has043 Martin [1990] Martin PP (1990) Representations of graph Temperley-Lieb algebras. Publ Res Inst Math Sci 26(3):485–503. 10.2977/prims/1195170958, URL https://doi.org/10.2977/prims/1195170958 Martin [2008] Martin S (2008) Schur algebras and representation theory, Cambridge Tracts in Mathematics, vol 112. Cambridge University Press, Cambridge, reprint of the 1993 original Munn [1955] Munn WD (1955) On semigroup algebras. Proc Cambridge Philos Soc 51:1–15 Munn [1957a] Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Martin P, Mazorchuk V (2014) On the representation theory of partial Brauer algebras. Q J Math 65(1):225–247. 10.1093/qmath/has043, URL https://doi.org/10.1093/qmath/has043 Martin [1990] Martin PP (1990) Representations of graph Temperley-Lieb algebras. Publ Res Inst Math Sci 26(3):485–503. 10.2977/prims/1195170958, URL https://doi.org/10.2977/prims/1195170958 Martin [2008] Martin S (2008) Schur algebras and representation theory, Cambridge Tracts in Mathematics, vol 112. Cambridge University Press, Cambridge, reprint of the 1993 original Munn [1955] Munn WD (1955) On semigroup algebras. Proc Cambridge Philos Soc 51:1–15 Munn [1957a] Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Martin PP (1990) Representations of graph Temperley-Lieb algebras. Publ Res Inst Math Sci 26(3):485–503. 10.2977/prims/1195170958, URL https://doi.org/10.2977/prims/1195170958 Martin [2008] Martin S (2008) Schur algebras and representation theory, Cambridge Tracts in Mathematics, vol 112. Cambridge University Press, Cambridge, reprint of the 1993 original Munn [1955] Munn WD (1955) On semigroup algebras. Proc Cambridge Philos Soc 51:1–15 Munn [1957a] Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Martin S (2008) Schur algebras and representation theory, Cambridge Tracts in Mathematics, vol 112. Cambridge University Press, Cambridge, reprint of the 1993 original Munn [1955] Munn WD (1955) On semigroup algebras. Proc Cambridge Philos Soc 51:1–15 Munn [1957a] Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Munn WD (1955) On semigroup algebras. Proc Cambridge Philos Soc 51:1–15 Munn [1957a] Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J.
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Publ Res Inst Math Sci 26(3):485–503. 10.2977/prims/1195170958, URL https://doi.org/10.2977/prims/1195170958 Martin [2008] Martin S (2008) Schur algebras and representation theory, Cambridge Tracts in Mathematics, vol 112. Cambridge University Press, Cambridge, reprint of the 1993 original Munn [1955] Munn WD (1955) On semigroup algebras. Proc Cambridge Philos Soc 51:1–15 Munn [1957a] Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. 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J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Martin P (1994) Temperley-Lieb algebras for nonplanar statistical mechanics—the partition algebra construction. J Knot Theory Ramifications 3(1):51–82. 10.1142/S0218216594000071, URL http://dx.doi.org/10.1142/S0218216594000071 Martin [1996] Martin P (1996) The structure of the partition algebras. J Algebra 183(2):319–358. 10.1006/jabr.1996.0223, URL http://dx.doi.org/10.1006/jabr.1996.0223 Martin and Mazorchuk [2014] Martin P, Mazorchuk V (2014) On the representation theory of partial Brauer algebras. Q J Math 65(1):225–247. 10.1093/qmath/has043, URL https://doi.org/10.1093/qmath/has043 Martin [1990] Martin PP (1990) Representations of graph Temperley-Lieb algebras. Publ Res Inst Math Sci 26(3):485–503. 10.2977/prims/1195170958, URL https://doi.org/10.2977/prims/1195170958 Martin [2008] Martin S (2008) Schur algebras and representation theory, Cambridge Tracts in Mathematics, vol 112. Cambridge University Press, Cambridge, reprint of the 1993 original Munn [1955] Munn WD (1955) On semigroup algebras. Proc Cambridge Philos Soc 51:1–15 Munn [1957a] Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Martin P (1996) The structure of the partition algebras. J Algebra 183(2):319–358. 10.1006/jabr.1996.0223, URL http://dx.doi.org/10.1006/jabr.1996.0223 Martin and Mazorchuk [2014] Martin P, Mazorchuk V (2014) On the representation theory of partial Brauer algebras. Q J Math 65(1):225–247. 10.1093/qmath/has043, URL https://doi.org/10.1093/qmath/has043 Martin [1990] Martin PP (1990) Representations of graph Temperley-Lieb algebras. Publ Res Inst Math Sci 26(3):485–503. 10.2977/prims/1195170958, URL https://doi.org/10.2977/prims/1195170958 Martin [2008] Martin S (2008) Schur algebras and representation theory, Cambridge Tracts in Mathematics, vol 112. Cambridge University Press, Cambridge, reprint of the 1993 original Munn [1955] Munn WD (1955) On semigroup algebras. Proc Cambridge Philos Soc 51:1–15 Munn [1957a] Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Martin P, Mazorchuk V (2014) On the representation theory of partial Brauer algebras. Q J Math 65(1):225–247. 10.1093/qmath/has043, URL https://doi.org/10.1093/qmath/has043 Martin [1990] Martin PP (1990) Representations of graph Temperley-Lieb algebras. Publ Res Inst Math Sci 26(3):485–503. 10.2977/prims/1195170958, URL https://doi.org/10.2977/prims/1195170958 Martin [2008] Martin S (2008) Schur algebras and representation theory, Cambridge Tracts in Mathematics, vol 112. Cambridge University Press, Cambridge, reprint of the 1993 original Munn [1955] Munn WD (1955) On semigroup algebras. Proc Cambridge Philos Soc 51:1–15 Munn [1957a] Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Martin PP (1990) Representations of graph Temperley-Lieb algebras. Publ Res Inst Math Sci 26(3):485–503. 10.2977/prims/1195170958, URL https://doi.org/10.2977/prims/1195170958 Martin [2008] Martin S (2008) Schur algebras and representation theory, Cambridge Tracts in Mathematics, vol 112. Cambridge University Press, Cambridge, reprint of the 1993 original Munn [1955] Munn WD (1955) On semigroup algebras. Proc Cambridge Philos Soc 51:1–15 Munn [1957a] Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Martin S (2008) Schur algebras and representation theory, Cambridge Tracts in Mathematics, vol 112. Cambridge University Press, Cambridge, reprint of the 1993 original Munn [1955] Munn WD (1955) On semigroup algebras. Proc Cambridge Philos Soc 51:1–15 Munn [1957a] Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Munn WD (1955) On semigroup algebras. Proc Cambridge Philos Soc 51:1–15 Munn [1957a] Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J.
- Martin P (1994) Temperley-Lieb algebras for nonplanar statistical mechanics—the partition algebra construction. J Knot Theory Ramifications 3(1):51–82. 10.1142/S0218216594000071, URL http://dx.doi.org/10.1142/S0218216594000071 Martin [1996] Martin P (1996) The structure of the partition algebras. J Algebra 183(2):319–358. 10.1006/jabr.1996.0223, URL http://dx.doi.org/10.1006/jabr.1996.0223 Martin and Mazorchuk [2014] Martin P, Mazorchuk V (2014) On the representation theory of partial Brauer algebras. Q J Math 65(1):225–247. 10.1093/qmath/has043, URL https://doi.org/10.1093/qmath/has043 Martin [1990] Martin PP (1990) Representations of graph Temperley-Lieb algebras. Publ Res Inst Math Sci 26(3):485–503. 10.2977/prims/1195170958, URL https://doi.org/10.2977/prims/1195170958 Martin [2008] Martin S (2008) Schur algebras and representation theory, Cambridge Tracts in Mathematics, vol 112. Cambridge University Press, Cambridge, reprint of the 1993 original Munn [1955] Munn WD (1955) On semigroup algebras. Proc Cambridge Philos Soc 51:1–15 Munn [1957a] Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Martin P (1996) The structure of the partition algebras. J Algebra 183(2):319–358. 10.1006/jabr.1996.0223, URL http://dx.doi.org/10.1006/jabr.1996.0223 Martin and Mazorchuk [2014] Martin P, Mazorchuk V (2014) On the representation theory of partial Brauer algebras. Q J Math 65(1):225–247. 10.1093/qmath/has043, URL https://doi.org/10.1093/qmath/has043 Martin [1990] Martin PP (1990) Representations of graph Temperley-Lieb algebras. Publ Res Inst Math Sci 26(3):485–503. 10.2977/prims/1195170958, URL https://doi.org/10.2977/prims/1195170958 Martin [2008] Martin S (2008) Schur algebras and representation theory, Cambridge Tracts in Mathematics, vol 112. Cambridge University Press, Cambridge, reprint of the 1993 original Munn [1955] Munn WD (1955) On semigroup algebras. Proc Cambridge Philos Soc 51:1–15 Munn [1957a] Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Martin P, Mazorchuk V (2014) On the representation theory of partial Brauer algebras. Q J Math 65(1):225–247. 10.1093/qmath/has043, URL https://doi.org/10.1093/qmath/has043 Martin [1990] Martin PP (1990) Representations of graph Temperley-Lieb algebras. Publ Res Inst Math Sci 26(3):485–503. 10.2977/prims/1195170958, URL https://doi.org/10.2977/prims/1195170958 Martin [2008] Martin S (2008) Schur algebras and representation theory, Cambridge Tracts in Mathematics, vol 112. Cambridge University Press, Cambridge, reprint of the 1993 original Munn [1955] Munn WD (1955) On semigroup algebras. Proc Cambridge Philos Soc 51:1–15 Munn [1957a] Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Martin PP (1990) Representations of graph Temperley-Lieb algebras. Publ Res Inst Math Sci 26(3):485–503. 10.2977/prims/1195170958, URL https://doi.org/10.2977/prims/1195170958 Martin [2008] Martin S (2008) Schur algebras and representation theory, Cambridge Tracts in Mathematics, vol 112. Cambridge University Press, Cambridge, reprint of the 1993 original Munn [1955] Munn WD (1955) On semigroup algebras. Proc Cambridge Philos Soc 51:1–15 Munn [1957a] Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Martin S (2008) Schur algebras and representation theory, Cambridge Tracts in Mathematics, vol 112. Cambridge University Press, Cambridge, reprint of the 1993 original Munn [1955] Munn WD (1955) On semigroup algebras. Proc Cambridge Philos Soc 51:1–15 Munn [1957a] Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Munn WD (1955) On semigroup algebras. Proc Cambridge Philos Soc 51:1–15 Munn [1957a] Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. 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Their Invariants and Representations. Princeton University Press, Princeton, N.J. Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J.
- Martin P (1996) The structure of the partition algebras. J Algebra 183(2):319–358. 10.1006/jabr.1996.0223, URL http://dx.doi.org/10.1006/jabr.1996.0223 Martin and Mazorchuk [2014] Martin P, Mazorchuk V (2014) On the representation theory of partial Brauer algebras. Q J Math 65(1):225–247. 10.1093/qmath/has043, URL https://doi.org/10.1093/qmath/has043 Martin [1990] Martin PP (1990) Representations of graph Temperley-Lieb algebras. Publ Res Inst Math Sci 26(3):485–503. 10.2977/prims/1195170958, URL https://doi.org/10.2977/prims/1195170958 Martin [2008] Martin S (2008) Schur algebras and representation theory, Cambridge Tracts in Mathematics, vol 112. Cambridge University Press, Cambridge, reprint of the 1993 original Munn [1955] Munn WD (1955) On semigroup algebras. Proc Cambridge Philos Soc 51:1–15 Munn [1957a] Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Martin P, Mazorchuk V (2014) On the representation theory of partial Brauer algebras. Q J Math 65(1):225–247. 10.1093/qmath/has043, URL https://doi.org/10.1093/qmath/has043 Martin [1990] Martin PP (1990) Representations of graph Temperley-Lieb algebras. Publ Res Inst Math Sci 26(3):485–503. 10.2977/prims/1195170958, URL https://doi.org/10.2977/prims/1195170958 Martin [2008] Martin S (2008) Schur algebras and representation theory, Cambridge Tracts in Mathematics, vol 112. Cambridge University Press, Cambridge, reprint of the 1993 original Munn [1955] Munn WD (1955) On semigroup algebras. Proc Cambridge Philos Soc 51:1–15 Munn [1957a] Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Martin PP (1990) Representations of graph Temperley-Lieb algebras. Publ Res Inst Math Sci 26(3):485–503. 10.2977/prims/1195170958, URL https://doi.org/10.2977/prims/1195170958 Martin [2008] Martin S (2008) Schur algebras and representation theory, Cambridge Tracts in Mathematics, vol 112. Cambridge University Press, Cambridge, reprint of the 1993 original Munn [1955] Munn WD (1955) On semigroup algebras. Proc Cambridge Philos Soc 51:1–15 Munn [1957a] Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Martin S (2008) Schur algebras and representation theory, Cambridge Tracts in Mathematics, vol 112. Cambridge University Press, Cambridge, reprint of the 1993 original Munn [1955] Munn WD (1955) On semigroup algebras. Proc Cambridge Philos Soc 51:1–15 Munn [1957a] Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Munn WD (1955) On semigroup algebras. Proc Cambridge Philos Soc 51:1–15 Munn [1957a] Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J.
- Martin P, Mazorchuk V (2014) On the representation theory of partial Brauer algebras. Q J Math 65(1):225–247. 10.1093/qmath/has043, URL https://doi.org/10.1093/qmath/has043 Martin [1990] Martin PP (1990) Representations of graph Temperley-Lieb algebras. Publ Res Inst Math Sci 26(3):485–503. 10.2977/prims/1195170958, URL https://doi.org/10.2977/prims/1195170958 Martin [2008] Martin S (2008) Schur algebras and representation theory, Cambridge Tracts in Mathematics, vol 112. Cambridge University Press, Cambridge, reprint of the 1993 original Munn [1955] Munn WD (1955) On semigroup algebras. Proc Cambridge Philos Soc 51:1–15 Munn [1957a] Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Martin PP (1990) Representations of graph Temperley-Lieb algebras. Publ Res Inst Math Sci 26(3):485–503. 10.2977/prims/1195170958, URL https://doi.org/10.2977/prims/1195170958 Martin [2008] Martin S (2008) Schur algebras and representation theory, Cambridge Tracts in Mathematics, vol 112. Cambridge University Press, Cambridge, reprint of the 1993 original Munn [1955] Munn WD (1955) On semigroup algebras. Proc Cambridge Philos Soc 51:1–15 Munn [1957a] Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Martin S (2008) Schur algebras and representation theory, Cambridge Tracts in Mathematics, vol 112. Cambridge University Press, Cambridge, reprint of the 1993 original Munn [1955] Munn WD (1955) On semigroup algebras. Proc Cambridge Philos Soc 51:1–15 Munn [1957a] Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Munn WD (1955) On semigroup algebras. Proc Cambridge Philos Soc 51:1–15 Munn [1957a] Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J.
- Martin PP (1990) Representations of graph Temperley-Lieb algebras. Publ Res Inst Math Sci 26(3):485–503. 10.2977/prims/1195170958, URL https://doi.org/10.2977/prims/1195170958 Martin [2008] Martin S (2008) Schur algebras and representation theory, Cambridge Tracts in Mathematics, vol 112. Cambridge University Press, Cambridge, reprint of the 1993 original Munn [1955] Munn WD (1955) On semigroup algebras. Proc Cambridge Philos Soc 51:1–15 Munn [1957a] Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. 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Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Martin S (2008) Schur algebras and representation theory, Cambridge Tracts in Mathematics, vol 112. Cambridge University Press, Cambridge, reprint of the 1993 original Munn [1955] Munn WD (1955) On semigroup algebras. Proc Cambridge Philos Soc 51:1–15 Munn [1957a] Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Munn WD (1955) On semigroup algebras. Proc Cambridge Philos Soc 51:1–15 Munn [1957a] Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. 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Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J.
- Martin S (2008) Schur algebras and representation theory, Cambridge Tracts in Mathematics, vol 112. Cambridge University Press, Cambridge, reprint of the 1993 original Munn [1955] Munn WD (1955) On semigroup algebras. Proc Cambridge Philos Soc 51:1–15 Munn [1957a] Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Munn WD (1955) On semigroup algebras. Proc Cambridge Philos Soc 51:1–15 Munn [1957a] Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J.
- Munn WD (1955) On semigroup algebras. Proc Cambridge Philos Soc 51:1–15 Munn [1957a] Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J.
- Munn WD (1957a) The characters of the symmetric inverse semigroup. Proc Cambridge Philos Soc 53:13–18 Munn [1957b] Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J.
- Munn WD (1957b) Matrix representations of semigroups. Proc Cambrdige Philos Soc 53:5–12 Paget [2006] Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J.
- Paget R (2006) Representation theory of q𝑞qitalic_q-rook monoid algebras. J Algebraic Combin 24(3):239–252. 10.1007/s10801-006-0010-y, URL http://dx.doi.org/10.1007/s10801-006-0010-y Ponizovskiĭ [1956] Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J.
- Ponizovskiĭ IS (1956) On matrix representations of associative systems. Mat Sb NS 38(80):241–260 Schur [1901] Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J.
- Schur I (1901) Über eine klasse von matrizen, die sich einer gegebenen matrix zuordnen lassen, dissertation. In I. Schur, Gesammelte Abhandlungen I, 1-70, Springer, Berlin, 1973 Schur [1927] Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J.
- Schur I (1927) Über die rationalen darstellungen der allgemeinen linearen gruppe. sitzber. königl. preuß. ak. wiss., physikal.-math. klasse, pages 58-75. In I. Schur, Gesammelte Abhandlungen. Band III, 68-85, Springer, Berlin, 1973. Solomon [1990] Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J.
- Solomon L (1990) The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36(1):15–49. 10.1007/BF00181463, URL http://dx.doi.org/10.1007/BF00181463 Solomon [2002] Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J.
- Solomon L (2002) Representations of the rook monoid. J Algebra 256(2):309–342. 10.1016/S0021-8693(02)00004-2, URL http://dx.doi.org/10.1016/S0021-8693(02)00004-2 Solomon [2004] Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J.
- Solomon L (2004) The Iwahori algebra of 𝕄n(𝔽q)subscript𝕄𝑛subscript𝔽𝑞\mathbb{M}_{n}(\mathbb{F}_{q})blackboard_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ). A presentation and a representation on tensor space. J Algebra 273(1):206–226. 10.1016/j.jalgebra.2003.08.013, URL http://dx.doi.org/10.1016/j.jalgebra.2003.08.013 Steinberg [2006] Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J.
- Steinberg B (2006) Möbius functions and semigroup representation theory. J Combin Theory Ser A 113(5):866–881. 10.1016/j.jcta.2005.08.004, URL http://dx.doi.org/10.1016/j.jcta.2005.08.004 Thrall [1944] Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J.
- Thrall RM (1944) On the decomposition of modular tensors. II. Ann of Math (2) 45:639–657. 10.2307/1969294, URL https://doi.org/10.2307/1969294 Vagner [1952] Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J.
- Vagner VV (1952) Generalized groups. Doklady Akad Nauk SSSR (NS) 84:1119–1122 Weyl [1939] Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J. Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J.
- Weyl H (1939) The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J.