Representations of finite matrix monoids

TL;DR

This paper provides explicit formulas for units in the monoid algebra of finite matrix monoids, describes simple modules, and establishes duality results, advancing the understanding of their algebraic structure.

Contribution

It introduces an explicit formula for the units spanning certain ideals and develops the module theory and duality for the monoid algebra of finite matrix monoids.

Findings

Explicit formula for units spanning ideals of matrices of rank at most r

Complete description of simple modules and their induction/restriction rules

Establishment of a Schur--Weyl duality for the monoid algebra

Abstract

Let $\mathfrak{M}_n$ be the multiplicative monoid of $n \times n$ matrices over a finite field. The monoid algebra $\mathbf{C}[\mathfrak{M}_n]$ has been studied for several decades. One of the important early results is Kov\'acs' theorem that the two-sided ideal spanned by matrices of rank at most $r$ has a unit. Our most significant result is an explicit formula for this unit. Prior to our work, such a formula was only known in a few examples. We also study the module theory of $\mathbf{C}[\mathfrak{M}_n]$. We explicitly describe the simple modules, and establish induction and restriction rules. We show that the simple decomposition of an arbitrary module can be determined using character theory of finite general linear groups; this relies on a Pieri rule of Gurevich--Howe. We also establish a version of Schur--Weyl duality for $\mathbf{C}[\mathfrak{M}_n]$. Many of these results hold over more general coefficient fields.