The Hilbert series of the superspace coinvariant ring

Let ${\mathrm{\Omega }}_n$ be the ring of polynomial-valued holomorphic differential forms on complex $n$-space, referred to in physics as the superspace ring of rank $n$. The symmetric group ${\mathfrak{S}}_n$ acts diagonally on ${\mathrm{\Omega }}_n$ by permuting commuting and anticommuting generators simultaneously. We let $S{I}_n\subseteq {\mathrm{\Omega }}_n$ be the ideal generated by ${\mathfrak{S}}_n$-invariants with vanishing constant term and study the quotient $S{R}_n={\mathrm{\Omega }}_n\mathrm{/}S{I}_n$ of superspace by this ideal. We calculate the doubly-graded Hilbert series of $S{R}_n$ and prove an `operator theorem’ which characterizes the harmonic space $S{H}_n\subseteq {\mathrm{\Omega }}_n$ attached to $S{R}_n$ in terms of the Vandermonde determinant and certain differential operators. Our methods employ commutative algebra results which were used in the study of Hessenberg varieties. Our results prove conjectures of N. Bergeron, Li, Machacek, Sulzgruber, Swanson, Wallach, and Zabrocki.