Quasiinvariants of $S_3$

Abstract

Let $s_{ij}$ represent a tranposition in $S_n$. A polynomial $P$ in $\mathbb{Q}[X_n]$ is said to be $m$-quasiinvariant with respect to $S_n$ if $(x_i-x_j)^{2m+1}$ divides $(1-s_{ij})P$ for all $1 \leq i, j \leq n$. We call the ring $m$-quasiinvariants $QI_m[X_n]$. We describe a method for constructing a basis for the quotient $QI_m[X_3]/< e_1, e_2, e_3>$. This leads to the evaluation of certain binomial determinants that are interesting in their own right.