Representations of finite matrix monoids

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.