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arXiv · 2404.17919

Superspace coinvariants and hyperplane arrangements

Robert Angarone·Patricia Commins·Trevor Karn·Satoshi Murai·Brendon Rhoades
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Abstract

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

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Robert Angarone, Patricia Commins, Trevor Karn, Satoshi Murai, Brendon Rhoades. 2026-01-07. Superspace coinvariants and hyperplane arrangements. https://arxiv.org/abs/2404.17919

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