The superspace coinvariant ring of type B

TL;DR

This paper studies the superspace coinvariant ring of type B, deriving its Hilbert series, describing the superharmonic space, and providing an explicit basis using hyperplane arrangements.

Contribution

It proves the conjectured Hilbert series and superharmonic space description for the superspace coinvariant ring of type B, and constructs an explicit basis.

Findings

Derived the Hilbert series of the superspace coinvariant ring of type B.

Proved the operator theorem describing the superharmonic space.

Constructed an explicit basis using hyperplane arrangements.

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.