Set superpartitions and superspace duality modules

2021/04/12 by Brendon Rhoades, Rhoades, Brendon, Andrew Timothy Wilson +1 · 3 citations
Computer Science · Mathematics · #Algebraic structures and combinatorial models #Coding theory and cryptography #Combinatorics (math.CO) #Commutative Algebra and Its Applications #FOS: Mathematics

paper · pdf · doi:10.48550/arxiv.2104.05630

openalex publication_date 2021/04/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

The superspace ring Ω_n is a rank n polynomial ring tensor a rank n exterior algebra. Using an extension of the Vandermonde determinant to Ω_n, the authors previously defined a family of doubly graded quotients \mathbbW_n,k of Ω_n which carry an action of the symmetric group \mathfrakS_n and satisfy a bigraded version of Poincaré Duality. In this paper, we examine the duality modules \mathbbW_n,k in greater detail. We describe a monomial basis of \mathbbW_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 (B₁ | ⋯ | B_k) of \1,…,n\ in which the non-minimal elements of any block B_i may be barred or unbarred.

Cited by

• Bosonic content of three-fermion high-spin states
• The superspace coinvariant ring of type B
• A proof of the Fields Conjectures

Related

• Vandermondes in superspace
• The superspace coinvariant ring of type B
• A proof of the Fields Conjectures
• Superspace coinvariants and hyperplane arrangements
• Ordered set partitions, generalized coinvariant algebras, and the Delta Conjecture