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Elementary methods for splitting representations of Rook monoids: a gentle introduction to groupoids (2410.23731v1)
Published 31 Oct 2024 in cs.DM
Abstract: We show that the algebra of the coloured rook monoid $R_n{(r)}$, {\em i.e.} the monoid of $n \times n$ matrices with at most one non-zero entry (an $r$-th root of unity) in each column and row, is the algebra of a finite groupoid, thus is endowed with a $C*$-algebra structure. This new perspective uncovers the representation theory of these monoid algebras by making manifest their decomposition in irreducible modules.