# Invariants of polynomials mod Frobenius powers

## TL;DR

This paper explores invariants of polynomials under Frobenius powers, providing an analog of Catalan number relations in positive characteristic and proving a variant of a conjecture for certain group actions.

## Contribution

It introduces a new analog in positive characteristic and proves a variant of a conjecture for finite general linear groups fixing a hyperplane.

## Key findings

- Proved a variant of Lewis-Reiner-Stanton conjecture in the local case.
- Established connections between invariants and Catalan numbers in positive characteristic.
- Extended understanding of polynomial invariants under Frobenius powers.

## Abstract

Lewis, Reiner, and Stanton conjectured a Hilbert seriesfor a space of invariants under an action of finite general linear groups using $(q,t)$-binomial coefficients. This work gives an analog in positive characteristic of theorems relating various Catalan numbers to the representation theory of rational Cherednik algebras. They consider a finite general linear group as a reflection group acting on the quotient of a polynomial ring by iterated powers of the irrelevant ideal under the Frobenius map. We prove a variant of their conjecture in the local case, when the group acting fixes a reflecting hyperplane.