Increasing subsequences, matrix loci, and Viennot shadows
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2023
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| _version_ | 1866910675511541760 |
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| author | Rhoades, Brendon |
| author_facet | Rhoades, Brendon |
| contents | Let $\mathbf{x}_{n \times n}$ be an $n \times n$ matrix of variables and let $\mathbb{F}[\mathbf{x}_{n \times n}]$ be the polynomial ring in these variables over a field $\mathbb{F}$. We study the ideal $I_n \subseteq \mathbb{F}[\mathbf{x}_{n \times n}]$ generated by all row and column variable sums and all products of two variables drawn from the same row or column. We show that the quotient $\mathbb{F}[\mathbf{x}_{n \times n}]/I_n$ admits a standard monomial basis determined by Viennot's shadow line avatar of the Schensted correspondence. As a corollary, the Hilbert series of $\mathbb{F}[\mathbf{x}_{n \times n}]/I_n$ is the generating function of permutations in $\mathfrak{S}_n$ by the length of their longest increasing subsequence. Along the way, we describe a `shadow junta' basis of the vector space of $k$-local permutation statistics. We also calculate the structure of $\mathbb{F}[\mathbf{x}_{n \times n}]/I_n$ as a graded $\mathfrak{S}_n \times \mathfrak{S}_n$-module. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2306_08718 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Increasing subsequences, matrix loci, and Viennot shadows Rhoades, Brendon Combinatorics Cryptography and Security Let $\mathbf{x}_{n \times n}$ be an $n \times n$ matrix of variables and let $\mathbb{F}[\mathbf{x}_{n \times n}]$ be the polynomial ring in these variables over a field $\mathbb{F}$. We study the ideal $I_n \subseteq \mathbb{F}[\mathbf{x}_{n \times n}]$ generated by all row and column variable sums and all products of two variables drawn from the same row or column. We show that the quotient $\mathbb{F}[\mathbf{x}_{n \times n}]/I_n$ admits a standard monomial basis determined by Viennot's shadow line avatar of the Schensted correspondence. As a corollary, the Hilbert series of $\mathbb{F}[\mathbf{x}_{n \times n}]/I_n$ is the generating function of permutations in $\mathfrak{S}_n$ by the length of their longest increasing subsequence. Along the way, we describe a `shadow junta' basis of the vector space of $k$-local permutation statistics. We also calculate the structure of $\mathbb{F}[\mathbf{x}_{n \times n}]/I_n$ as a graded $\mathfrak{S}_n \times \mathfrak{S}_n$-module. |
| title | Increasing subsequences, matrix loci, and Viennot shadows |
| topic | Combinatorics Cryptography and Security |
| url | https://arxiv.org/abs/2306.08718 |