Let $M_n = M(n, F_p)$ be the semigroup of all $n\times n$ matrices over the field $F_p$ of $p$ elements, $p$ a prime number. As well known, each irreducible $M_n$-module appears as a composition factor of the space of homogeneous polynomials in some degree $d$. The purpose of the paper is to determine the lowest degree $d$ for some irreducibles modules.