Derangement permutation matrices and orbit harmonics

Abstract

Let ${\mathbf{x}}_{n\times n}$ be an $n\times n$ matrix of variables and let $S=\mathbb{F}[{\mathbf{x}}_{n\times n}]$ be the polynomial ring over these variables where $\mathbb{F}$ is a field of characteristic zero. Regard $S$ as the coordinate ring of the affine space ${\mathbb{F}}^{n\times n}$ of $n\times n$ $\mathbb{F}$-matrices. Let ${\mathfrak{D}}_n\subseteq {\mathbb{F}}^{n\times n}$ be the locus of derangement permutation matrices. We study the orbit harmonics quotient ring $\mathbf{R}({\mathfrak{D}}_n)=S/\mathrm{gr}\mathbf{I}({\mathfrak{D}}_n)$ where $\mathrm{gr}\mathbf{I}({\mathfrak{D}}_n)$ is the associated graded ideal of the vanishing ideal $\mathbf{I}({\mathfrak{D}}_n)\subseteq S$. We give an explicit generating set of $\mathrm{gr}\mathbf{I}({\mathfrak{D}}_n),$ relate the Hilbert series of $\mathbf{R}({\mathfrak{D}}_n)$ to the Foata transformation and the longest increasing subsequence statistic on ${\mathfrak{S}}_n$, and give an alternating sum formula for the graded ${\mathfrak{S}}_n$-character of $\mathbf{R}({\mathfrak{D}}_n)$. Our proofs make heavy use of the mapping cone construction of homological algebra.

34 pages

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