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

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.