Superspace coinvariants and hyperplane arrangements

TL;DR

This paper constructs an explicit basis for the superspace coinvariant ring using hyperplane arrangements, confirming a conjecture and connecting algebraic and combinatorial structures.

Contribution

It provides the first explicit basis for the superspace coinvariant ring, solving a conjecture and employing hyperplane arrangement techniques.

Findings

Explicit basis of superspace coinvariant ring constructed

Connection established between coinvariant rings and Solomon-Terao algebras

Techniques involve derivation modules of specific hyperplane arrangements

Abstract

Let $\Omega$ be the {\em superspace ring} of polynomial-valued differential forms on affine $n$-space. The natural action of the symmetric group $\mathfrak{S}_n$ on $n$-space induces an action of $\mathfrak{S}_n$ on $\Omega$. The {\em superspace coinvariant ring} is the quotient $SR$ of $\Omega$ by the ideal generated by $\mathfrak{S}_n$-invariants with vanishing constant term. We give the first explicit basis of $SR$, proving a conjecture of Sagan and Swanson. Our techniques use the theory of hyperplane arrangements. We relate $SR$ to instances of the Solomon-Terao algebras of Abe-Maeno-Murai-Numata and use exact sequences relating the derivation modules of certain `southwest closed' arrangements to obtain the desired basis of $SR$.