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

Abstract The superspace ring $$\Omega _n$$ is a rank n polynomial ring tensored with 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 Poincaré 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 ordered set superpartitions . These are ordered set partitions $$(B_1 \mid \cdots \mid B_k)$$ of $$\{1,\dots ,n\}$$ in which the nonminimal elements of any block $$B_i$$ may be barred or unbarred.