The superspace coinvariant ring of type B

Abstract

Given the rank $n$ superspace $\Omega_n$, 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 $SR_n^B$, defined as the quotient of $\Omega_n$ by two-sided ideal generated by all $\mathfrak B_n$ invariants with vanishing constant terms. We derive the Hilbert series of $SR^B_n$ conjectured by Sagan and Swanson, and prove an operator theorem that yields a concrete description of the superharmonic space $SH^B_n$ associated to $SR^B_n$ as conjectured by Swanson and Wallach. We also derive an explicit basis of $SR^B_n$ using the theory of hyperplane arrangements.