The Hilbert series of the superspace coinvariant ring
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866909395070222336 |
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| author | Rhoades, Brendon Wilson, Andy |
| author_facet | Rhoades, Brendon Wilson, Andy |
| contents | Let $Ω_n$ be the ring of polynomial-valued holomorphic differential forms on complex $n$-space, referred to in physics as the superspace ring of rank $n$. The symmetric group $\mathfrak{S}_n$ acts diagonally on $Ω_n$ by permuting commuting and anticommuting generators simultaneously. We let $SI_n \subseteq Ω_n$ be the ideal generated by $\mathfrak{S}_n$-invariants with vanishing constant term and study the quotient $SR_n = Ω_n / SI_n$ of superspace by this ideal. We calculate the doubly-graded Hilbert series of $SR_n$ and prove an `operator theorem' which characterizes the harmonic space $SH_n \subseteq Ω_n$ attached to $SR_n$ in terms of the Vandermonde determinant and certain differential operators. Our methods employ commutative algebra results which were used in the study of Hessenberg varieties. Our results prove conjectures of N. Bergeron, Li, Machacek, Sulzgruber, Swanson, Wallach, and Zabrocki. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2301_09763 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | The Hilbert series of the superspace coinvariant ring Rhoades, Brendon Wilson, Andy Combinatorics Let $Ω_n$ be the ring of polynomial-valued holomorphic differential forms on complex $n$-space, referred to in physics as the superspace ring of rank $n$. The symmetric group $\mathfrak{S}_n$ acts diagonally on $Ω_n$ by permuting commuting and anticommuting generators simultaneously. We let $SI_n \subseteq Ω_n$ be the ideal generated by $\mathfrak{S}_n$-invariants with vanishing constant term and study the quotient $SR_n = Ω_n / SI_n$ of superspace by this ideal. We calculate the doubly-graded Hilbert series of $SR_n$ and prove an `operator theorem' which characterizes the harmonic space $SH_n \subseteq Ω_n$ attached to $SR_n$ in terms of the Vandermonde determinant and certain differential operators. Our methods employ commutative algebra results which were used in the study of Hessenberg varieties. Our results prove conjectures of N. Bergeron, Li, Machacek, Sulzgruber, Swanson, Wallach, and Zabrocki. |
| title | The Hilbert series of the superspace coinvariant ring |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2301.09763 |