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Title: Set superpartitions and superspace duality modules
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.  more » « less
Award ID(s):
1953781
PAR ID:
10432002
Author(s) / Creator(s):
;
Date Published:
Journal Name:
Forum of Mathematics, Sigma
Volume:
10
ISSN:
2050-5094
Format(s):
Medium: X
Sponsoring Org:
National Science Foundation
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