Superspace coinvariants and hyperplane arrangements

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