Invariant Theory of finite general linear groups modulo Frobenius powers

Abstract

We prove some cases of a conjecture of Lewis, Reiner and Stanton regarding Hilbert series corresponding to the action of $G{l}_n({\mathbb{F}}_q)$ on a polynomial ring modulo Frobenius powers. We also give a few conjectures about the invariant ring for certain cases that we don't prove completely.