Hilbert Series of $S_3$-Quasi-Invariant Polynomials in Characteristics 2, 3

TL;DR

This paper computes the Hilbert series of quasi-invariant polynomials in three variables over fields of characteristic 2 and 3, providing explicit generators in characteristic 2 and extending previous work on modular representations.

Contribution

It extends the understanding of quasi-invariant polynomials in small characteristics, explicitly describes generators in characteristic 2, and establishes necessary and sufficient conditions for Hilbert series differences across characteristics.

Findings

Computed Hilbert series for n=3 in characteristics 2 and 3.

Explicitly described generators in characteristic 2.

Proved the necessary and sufficient condition for Hilbert series differences.

Abstract

We compute the Hilbert series of the space of $n=3$ variable quasi-invariant polynomials in characteristic $2$ and $3$, capturing the dimension of the homogeneous components of the space, and explicitly describe the generators in the characteristic $2$ case. In doing so we extend the work of the first author in 2023 on quasi-invariant polynomials in characteristic $p>n$ and prove that a sufficient condition found by Ren-Xu in 2020 on when the Hilbert series differs between characteristic $0$ and $p$ is also necessary for $n=3$, $p=2,3$. This is the first description of quasi-invariant polynomials in the case, where the space forms a modular representation over the symmetric group, bringing us closer to describing the quasi-invariant polynomials in all characteristics and numbers of variables.