Hilbert series of quasi-invariant polynomials in characteristics $p\leq n$

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.