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

TL;DR

This paper introduces a novel perspective on the algebra of coloured rook monoids by demonstrating its structure as a finite groupoid algebra, which clarifies their representation theory and module decomposition.

Contribution

It shows that the algebra of coloured rook monoids can be viewed as a finite groupoid algebra, providing new insights into their representation theory.

Findings

The algebra of coloured rook monoids has a $C^*$-algebra structure.

Representation theory is clarified through the decomposition into irreducible modules.

The approach offers a gentle introduction to the algebraic structure of rook monoids.

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.