The superspace coinvariant ring of type B

Abstract

Given the rank $n$ superspace $\hat{I}{\text{c◯}}_n$, the ring of polynomial-valued differential forms on ${\mathbb{C}}^n$, one can define an action of hyperoctahedral group ${\mathfrak{B}}_n$ on it. This leads to a superspace coinvariant ideal $S{R}_n^B$, defined as the quotient of $\hat{I}{\text{c◯}}_n$ by two-sided ideal generated by all ${\mathfrak{B}}_n$ invariants with vanishing constant terms. We derive the Hilbert series of $S{R}_n^B$ conjectured by Sagan and Swanson, and prove an operator theorem that yields a concrete description of the superharmonic space $S{H}_n^B$ associated to $S{R}_n^B$ as conjectured by Swanson and Wallach. We also derive an explicit basis of $S{R}_n^B$ using the theory of hyperplane arrangements.

26 pages