Increasing subsequences, matrix loci and Viennot shadows - Rhoades, Brendon (Forum of Mathematics, Sigma, 2024)
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.