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arXiv · 2410.23731

Elementary methods for splitting representations of Rook monoids: a gentle introduction to groupoids

Gérard Henry Edmond Duchamp·Joseph Ben Geloun·Christophe Tollu
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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.

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Gérard Henry Edmond Duchamp, Joseph Ben Geloun, Christophe Tollu. 2024-10-31. Elementary methods for splitting representations of Rook monoids: a gentle introduction to groupoids. https://arxiv.org/abs/2410.23731

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