Representations of finite matrix monoids
Nate Harman, Andrew Snowden, Elad Zelingher
TL;DR
The paper analyzes the monoid algebra k[M_n] of n×n matrices over a finite field, obtaining an explicit unit for Kovács’ rank r ideals and developing a detailed module theory that classifies simple modules L_n(π) and their induction/restriction behavior. It introduces semi-idempotent matrices and a robust counting framework that yields the unit formula, built on q-binomial coefficients and stable rank, and leverages groupoid algebras to connect the monoid structure with GL representations. A Pieri-type rule for finite general linear groups and a Schur-Weyl duality for k[M_n] are established, enabling explicit decompositions of tensor powers and a multiplicity formula from group-theoretic data. The work provides a solid foundation for applying these algebras in tensor categories and Jacobi categories, and offers a complete, calculable picture of the representation theory of matrix monoids over finite fields with broad conceptual and computational implications.
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 Kovács' 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.