Set superpartitions and superspace duality modules

TL;DR

This paper explores duality modules in superspace rings, introducing combinatorial objects called ordered superpartitions to describe their structure and properties.

Contribution

It provides a detailed analysis of duality modules in superspace, including a monomial basis and combinatorial formulas for their bigraded Hilbert and Frobenius series.

Findings

Established a monomial basis for the modules.

Derived combinatorial formulas involving ordered superpartitions.

Connected algebraic structures with new combinatorial objects.

Abstract

The superspace ring $\Omega_n$ is a rank $n$ polynomial ring tensor a rank $n$ exterior algebra. Using an extension of the Vandermonde determinant to $\Omega_n$, the authors previously defined a family of doubly graded quotients $\mathbb{W}_{n,k}$ of $\Omega_n$ which carry an action of the symmetric group $\mathfrak{S}_n$ and satisfy a bigraded version of Poincar\'e Duality. In this paper, we examine the duality modules $\mathbb{W}_{n,k}$ in greater detail. We describe a monomial basis of $\mathbb{W}_{n,k}$ and give combinatorial formulas for its bigraded Hilbert and Frobenius series. These formulas involve new combinatorial objects called {\em ordered superpartitions}. These are ordered set partitions $(B_1 \mid \cdots \mid B_k)$ of $\{1,\dots,n\}$ in which the non-minimal elements of any block $B_i$ may be barred or unbarred.