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| Auteurs principaux: | , , , , |
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| Format: | Preprint |
| Publié: |
2024
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| Accès en ligne: | https://arxiv.org/abs/2404.17919 |
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| _version_ | 1866911358115643392 |
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| author | Angarone, Robert Commins, Patricia Karn, Trevor Murai, Satoshi Rhoades, Brendon |
| author_facet | Angarone, Robert Commins, Patricia Karn, Trevor Murai, Satoshi Rhoades, Brendon |
| contents | 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 $SR$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_17919 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Superspace coinvariants and hyperplane arrangements Angarone, Robert Commins, Patricia Karn, Trevor Murai, Satoshi Rhoades, Brendon Combinatorics Commutative Algebra 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 $SR$. |
| title | Superspace coinvariants and hyperplane arrangements |
| topic | Combinatorics Commutative Algebra |
| url | https://arxiv.org/abs/2404.17919 |