On the first occurrence of irreducible modular representations of semigroups of all matrices (Q1808903)

The paper is concerned with irreducible modules over the semigroup algebra \({\mathbf F}_p[M_n]\) of the monoid \(M_n\) of all \(n\times n\) matrices over the prime field \({\mathbf F}_p\). It has been known that every such module embeds into some \(S^d\), with \(d\leq\sum^n_{i=1}(p^i-1)\), where \(S^d\) stands for the subspace of \({\mathbf F}_p[x_1,\dots,x_n]\) consisting of all homogeneous polynomials of degree \(d\), with the natural action of \(M_n\) [see \textit{S. Doty} and \textit{G. Walker}, Math. Proc. Camb. Philos. Soc. 119, No. 2, 231-242 (1996; Zbl 0855.20008)]. A complete list of irreducible \({\mathbf F}_p[M_n]\)-modules \(H_\beta\) was given by the author [in Acta. Math. Vietnam 20, No. 1, 43-53 (1995; Zbl 0886.20039)]. Each \(H_\beta\) is a submodule of some \(S^d\). In this paper it is shown that this occurrence of \(H_\beta\) is the first occurrence of this module as a submodule of \({\mathbf F}_p[x_1,\dots,x_n]\).