Increasing subsequences, matrix loci, and Viennot shadows

• URL: http://arxiv.org/abs/2306.08718v2
• Date: Tue, 29 Oct 2024 01:21:04 GMT
• Title: Increasing subsequences, matrix loci, and Viennot shadows
• Authors: Brendon Rhoades,
• Abstract summary: We show that the quotient $mathbbF[mathbfx_n times n]/I_n$ admits a standard monomial basis. We also calculate the structure of $mathbbF[mathbfx_n times n]/I_n$ as a graded $mathfrakS_n times mathfrakS_n$-module.
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• Abstract: 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.

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