Superspace coinvariants and hyperplane arrangements

Abstract

Let Ω be the superspace ring of polynomial-valued differential forms on affine n-space. The natural action of the symmetric group $S_n$ on n-space induces an action of $S_n$ on Ω. The superspace coinvariant ring is the quotient SR of Ω by the ideal generated by $S_n$-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.