Increasing subsequences, matrix loci, and Viennot shadows

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.