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

Gérard Henry Edmond Duchamp, Joseph Ben Geloun, Christophe Tollu

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.

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

Abstract

We show that the algebra of the coloured rook monoid , {\em i.e.} the monoid of matrices with at most one non-zero entry (an -th root of unity) in each column and row, is the algebra of a finite groupoid, thus is endowed with a -algebra structure. This new perspective uncovers the representation theory of these monoid algebras by making manifest their decomposition in irreducible modules.

Paper Structure

This paper contains 8 sections, 3 theorems, 23 equations.

Table of Contents

  1. Introduction
  2. The rook monoids $R_n$ and coloured rook monoids $R_n^{(r)}$.
  3. Definitions
  4. Partial identities: decoupling ${\mathbb C}[R_n^{(r)}]$
  5. Generalities on groupoids
  6. The $C^*$-algebra of a finite groupoid and the rook monoid algebra
  7. Wedderburn-Artin decomposition of the algebra of a finite groupoid ${\mathbb C}[G]$ and of ${\mathbb C}[R_n^{(r)}]$
  8. Concluding remarks

Key Result

Proposition 1

We have the following i) $\{e_A\}_{A\subset [1,\ldots,n]}$ is a complete set of orthogonal idempotents of ${\mathbb Z}[D_n]$, i.e.$\sum_{A\subset[1,\ldots,n]} = 1_{\mathbb{Z}[D_n]}$ and $e_A e_B =\delta_{A,B} e_A$. ii) For all $M\in R_n^{(r)}$, iii) For $M,N\in R_n^{(r)}$, one has (iv) $\mathbf{1}_*:= \mu^{-1}([I_{n\times n}]) = \sum_{S\subset [1,\ldots, n]} I_S$ is the unit for the product $\as

Theorems & Definitions (3)

  • Proposition 1
  • Theorem 2
  • Lemma 3