Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 21 (2025), 057, 24 pages      arXiv:2412.20673      https://doi.org/10.3842/SIGMA.2025.057

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

Frank Wang and Eric Yee
Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA

Received January 31, 2025, in final form July 02, 2025; Published online July 13, 2025

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.

Key words: quasi-invariant polynomials; modular representation theory.

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