Involution matrix loci and orbit harmonics
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arXiv
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| Main Authors: | , , , |
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| Format: | Preprint |
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2024
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| _version_ | 1866917772252938240 |
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| author | Liu, Moxuan J. Ma, Yichen Rhoades, Brendon Zhu, Hai |
| author_facet | Liu, Moxuan J. Ma, Yichen Rhoades, Brendon Zhu, Hai |
| contents | Let $\mathrm{Mat}_{n \times n}(\mathbb{C})$ be the affine space of $n \times n$ complex matrices with coordinate ring $\mathbb{C}[\mathbf{x}_{n \times n}]$. We define graded quotients of $\mathbb{C}[\mathbf{x}_{n \times n}]$ which carry an action of the symmetric group $\mathfrak{S}_n$ by simultaneous permutation of rows and columns. These quotient rings are obtained by applying the orbit harmonics method to matrix loci corresponding to all involutions in $\mathfrak{S}_n$ and the conjugacy classes of involutions in $\mathfrak{S}_n$ with a given number of fixed points. In the case of perfect matchings on $\{1, \dots, n\}$ with $n$ even, the Hilbert series of our quotient ring is related to Tracy-Widom distributions and its graded Frobenius image gives a refinement of the plethysm $s_{n/2}[s_2]$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_06175 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Involution matrix loci and orbit harmonics Liu, Moxuan J. Ma, Yichen Rhoades, Brendon Zhu, Hai Combinatorics Let $\mathrm{Mat}_{n \times n}(\mathbb{C})$ be the affine space of $n \times n$ complex matrices with coordinate ring $\mathbb{C}[\mathbf{x}_{n \times n}]$. We define graded quotients of $\mathbb{C}[\mathbf{x}_{n \times n}]$ which carry an action of the symmetric group $\mathfrak{S}_n$ by simultaneous permutation of rows and columns. These quotient rings are obtained by applying the orbit harmonics method to matrix loci corresponding to all involutions in $\mathfrak{S}_n$ and the conjugacy classes of involutions in $\mathfrak{S}_n$ with a given number of fixed points. In the case of perfect matchings on $\{1, \dots, n\}$ with $n$ even, the Hilbert series of our quotient ring is related to Tracy-Widom distributions and its graded Frobenius image gives a refinement of the plethysm $s_{n/2}[s_2]$. |
| title | Involution matrix loci and orbit harmonics |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2409.06175 |