A proof of the Lewis-Reiner-Stanton conjecture for the Borel subgroup

TL;DR

This paper proves a conjecture about the Hilbert series of invariants under the Borel subgroup of GL_n over finite fields, by constructing an explicit basis using a novel operator.

Contribution

It provides the first proof of the Lewis-Reiner-Stanton conjecture for the Borel subgroup and introduces an operator-based method to construct invariants.

Findings

Confirmed the conjecture for the Borel subgroup.

Constructed an explicit basis for invariants.

Proposed a basis for all parabolic subgroups.

Abstract

For each parabolic subgroup $P$ of the general linear group $GL_n(\mathbb{F}_q)$, a conjecture due to Lewis, Reiner and Stanton \cite{LewisReinerStanton2017} predicts a formula for the Hilbert series of the space of invariants $\mathcal{Q}_m(n)^{P}$ where $\mathcal{Q}_m(n)$ is the quotient ring $\mathbb{F}_q[x_1,\ldots,x_n]/(x_1^{q^m},\ldots,x_n^{q^m})$. In this paper, we prove the conjecture for the Borel subgroup $B$ by constructing a linear basis for $mathcal{Q}_m(n)^B$. The construction is based on an operator $\delta$ which produces new invariants from old invariants of lower ranks. We also upgrade the conjecture of Lewis, Reiner and Stanton by proposing an explicit basis for the space of invariants for each parabolic subgroup.