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.

翻译：设 $\mathbf{x}_{n \times n}$ 为一个 $n \times n$ 变量矩阵，$\mathbb{F}[\mathbf{x}_{n \times n}]$ 为域 $\mathbb{F}$ 上由这些变量生成的多项式环。我们研究由所有行和与列和的变量和，以及所有取自同一行或同一列的两个变量乘积生成的理想 $I_n \subseteq \mathbb{F}[\mathbf{x}_{n \times n}]$。我们证明商环 $\mathbb{F}[\mathbf{x}_{n \times n}]/I_n$ 具有一个由维诺阴影线（基于申斯特德对应）确定的标准单项式基。作为推论，$\mathbb{F}[\mathbf{x}_{n \times n}]/I_n$ 的希尔伯特级数是 $\mathfrak{S}_n$ 中置换按最长递增子序列长度的生成函数。在此过程中，我们描述了 $k$-局域置换统计量向量空间的“阴影长老”基。同时，我们计算了 $\mathbb{F}[\mathbf{x}_{n \times n}]/I_n$ 作为分次 $\mathfrak{S}_n \times \mathfrak{S}_n$-模的结构。