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Galois descent of splendid Rickard equivalences for blocks of $p$-nilpotent groups (2405.16061v2)
Published 25 May 2024 in math.GR, math.RA, and math.RT
Abstract: We strengthen the results of Boltje and Yilmaz regarding the Galois descent of equivalences of blocks of $p$-nilpotent groups and a result of Kessar and Linckelmann regarding Galois descent of splendid Rickard equivalences for blocks with compatible Galois stabilizers. A more general descent criteria for chain complexes is proven along the way, which requires the adaptation of a theorem of Reiner for chain complexes. This verifies Kessar and Linckelmann's refinement of Brou\'{e}'s abelian defect group conjecture for blocks of $p$-nilpotent groups with abelian Sylow $p$-subgroup.
- R. Boltje and B. Xu. On p𝑝pitalic_p-permutation equivalences: betwen Rickard equivalences and isotypies. Transations of the American Mathematical Society, 360(10):5067–5087, 2008.
- R. Boltje and D. Yilmaz. Galois descent of equivalences between blocks of p𝑝pitalic_p-nilpotent groups. Proceedings of the American Mathematical Society, 150(2):559–573, 2022.
- J. Chuang and R. Rouquier. Derived equivalences for symmetric groups and 𝔰𝔩2𝔰subscript𝔩2\mathfrak{sl}_{2}fraktur_s fraktur_l start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT-categorification. Annals of Mathematics, 167:245–298, 2008.
- C. W. Curtis and I. Reiner. Methods of Representation Theory with Applications to Finite Groups & Orders, Volume 1. John Wiley & Sons, 1981.
- M. Harris and M. Linckelmann. Splendid derived equivalences for blocks of finite p𝑝pitalic_p-solvable groups. Journal of the London Mathematical Society, pages 85–96, 2000.
- D. Benson (https://mathoverflow.net/users/460592/dave benson). Extension of scalars for bounded chain complexes of kg𝑘𝑔kgitalic_k italic_g-modules. MathOverflow. URL:https://mathoverflow.net/q/471654 (version: 2024-05-20).
- X. Huang. Descent of equivalences for blocks with klein four defect groups. J. of Algebra, 614:898–905, 2023.
- The strengthened broué abelian defect group conjecture for SL(2,pn)SL2superscript𝑝𝑛\operatorname{SL}(2,p^{n})roman_SL ( 2 , italic_p start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT ) and GL(2,pn)GL2superscript𝑝𝑛\operatorname{GL}(2,p^{n})roman_GL ( 2 , italic_p start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT ). J. of Algebra, 633:114–137, 2023.
- R. Kessar and M. Linckelmann. Descent of equivalences and character bijections. In Geometric and topological aspects of the representation theory of finite groups, PROMS, pages 181–212. Springer, 2018.
- M. Linckelmann. The Block Theory of Finite Group Algebras, Volume 1. Cambridge University Press, 2018.
- M. Linckelmann. The Block Theory of Finite Group Algebras, Volume 2. Cambridge University Press, 2018.
- I. Reiner. Relations between integral and modular representations. Michigan Math J., 13:357–372, 1966.
- J. Rickard. Splendid equivalences: Derived categories and permutation modules. Proceedings of the London Mathematical Society, 72(2):331–358, 1996.
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