The first occurrence for the irreducible modules of general linear groups in the polynomial algebra

Let $p$ be a prime number and let $GL_{n}$ be the group of all invertible matrices over the prime field $\mathbb{F}_p$. It is known that every irreducible $GL_{n}$-module can occur as a submodule of $P$, the polynomial algebra with $n$ variables over $\mathbb{F}_p$. Given an irreducible $GL_{n}$-module $\rho$, the purpose of this paper is to find out the first value of the degree $d$ of which $\rho$ occurs as a submodule of $P_{d}$, the subset of $P$ consisting of homogeneous polynomials of degree $d$. This generalizes Schwartz-Tri's result to the case of any prime $p$.