错位排列置换矩阵与轨道调和

中文摘要 AI 辅助

设$\boldsymbol{x}_{n \times n}$为$n \times n$的变量矩阵，令$S = \boldsymbol{F}[\boldsymbol{x}_{n \times n}]$为这些变量上的多项式环，其中$\boldsymbol{F}$是特征为零的域。将$S$视为$n \times n$ $\boldsymbol{F}$矩阵构成的仿射空间$\boldsymbol{F}^{n \times n}$的坐标环。设$\boldsymbol{D}_n \boldsymbol{F}^{n \times n}$为错位排列置换矩阵的轨迹。我们研究轨道调和商环$\boldsymbol{R}(\boldsymbol{D}_n) = S/\text{gr}\boldsymbol{I}(\boldsymbol{D}_n)$，其中$\text{gr}\boldsymbol{I}(\boldsymbol{D}_n)$是消失理想$\boldsymbol{I}(\boldsymbol{D}_n) \boldsymbol{S}$的相伴分次理想。我们给出$\text{gr}\boldsymbol{I}(\boldsymbol{D}_n)$的显式生成元集合，将$\boldsymbol{R}(\boldsymbol{D}_n)$的希尔伯特级数与Foata变换及$\boldsymbol{\frak{S}}_n$上的最长递增子序列统计量关联起来，并给出$\boldsymbol{R}(\boldsymbol{D}_n)$的分次$\boldsymbol{\frak{S}}_n$特征的交替和公式。我们的证明大量使用同调代数的映射锥构造。

英文摘要

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 ${\bf R}(\mathfrak{D}_n) = S/\mathrm{gr} \, \mathbf{I}(\mathfrak{D}_n)$ 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.

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34 pages