Superspace coinvariants and hyperplane arrangements

Let Omega be the superspace ring of polynomial-valued differential forms on affine n-space. The natural action of the symmetric group enon n-space induces an action of enon Omega. The superspace coinvariant ring is the quotient SR of Omega by the ideal generated by en-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. (c) 2025 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.