Nguyen Van Nghia | Searcharxiv

arXiv2025

On Modular Invariants of Truncated Polynomial Rings in low ranks

We verify the conjectures due to Lewis, Reiner and Stanton about the Hilbert series of the invariant ring of the truncated polynomial ring for all parabolic subgroups up to rank $3$. This is done by constructing an explicit set of generators for each invariant ring in question. We also propose a conjecture concerning the action of the Steenrod algebra and the Dickson algebra on a certain naturally occurring filtration of the invariant ring under the general linear group.

math.RA↗

arXiv2024

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

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 $δ$ 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.

math.RA↗