On the first occurrence of irreducible representations of the matrix semigroup (Q700255)

Let \(M_n:=M(n,\mathbb F_p)\) be the semigroup of all \(n\times n\) matrices over the finite field \(\mathbb F_p\) of \(p\) elements, where \(p\) is a prime. It is well known that each irreducible \(M_n\)-module appears as a composition factor of the space of homogeneous polynomials in some degree \(d\). The article determines the lowest degree \(d\) for some irreducible modules, which generalizes results by \textit{D. P. Carlisle} and \textit{N. J. Kuhn} [J. Algebra 121, No. 2, 370--387 (1989; Zbl 0691.55015)] for \(p=2\).