Superspace coinvariants and hyperplane arrangements

Abstract

Let $\hat{I}\text{c◯}$ 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 $\hat{I}\text{c◯}$. The {\em superspace coinvariant ring} is the quotient $SR$ of $\hat{I}\text{c◯}$ 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 $SR$.

26 pages. This version organizes the paper as in the journal version (the former Section 3 is now an appendix). Also, an erroneous equivalent condition was removed from Proposition 4.4 (formerly Proposition 5.4); other results in the paper are unaffected