Superspace coinvariants and hyperplane arrangements

Let $\mathrm{\Omega }$ 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 $\mathrm{\Omega }$. The {\em superspace coinvariant ring} is the quotient $SR$ of $\mathrm{\Omega }$ 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 $SR$.