Hilbert Series of 𝑆₃-Quasi-Invariant Polynomials in Characteristics 2, 3

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.