On Modular Invariants of Truncated Polynomial Rings in low ranks

TL;DR

This paper verifies conjectures about the Hilbert series of invariant rings of truncated polynomial rings for low ranks by explicitly constructing generators, and proposes a new conjecture on algebra actions.

Contribution

It confirms existing conjectures for ranks up to 3 and introduces a new conjecture on Steenrod and Dickson algebra actions on invariant rings.

Findings

Verified conjectures for ranks up to 3

Constructed explicit generators for invariant rings

Proposed a new conjecture on algebra actions

Abstract

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.