We introduce a family of ideals $I_{n,\lambda,s}$ in $\mathbb{Q}[x_1,\dots,x_n]$ for $\lambda$ a partition of $k\leq n$ and an integer $s \geq \ell(\lambda)$. This family contains both the Tanisaki ideals $I_\lambda$ and the ideals $I_{n,k}$ of Haglund-Rhoades-Shimozono as special cases. We study the corresponding quotient rings $R_{n,\lambda,s}$ as symmetric group modules. When $n=k$ and $s$ is arbitrary, we recover the Garsia-Procesi modules, and when $\lambda=(1^k)$ and $s=k$, we recover the generalized coinvariant algebras of Haglund-Rhoades-Shimozono. We give a monomial basis for $R_{n,\lambda,s}$, unifying the monomial bases studied by Garsia-Procesi and Haglund-Rhoades-Shimozono, and realize the $S_n$-module structure of $R_{n,\lambda,s}$ in terms of an action on $(n,\lambda,s)$-ordered set partitions. We also prove formulas for the Hilbert series and graded Frobenius characteristic of $R_{n,\lambda,s}$. We then connect our work with Eisenbud-Saltman rank varieties using results of Weyman. As an application of our work, we give a monomial basis, Hilbert series formula, and graded Frobenius characteristic formula for the coordinate ring of the scheme-theoretic intersection of a rank variety with diagonal matrices.

中文翻译：

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有序集分区、Garsia-Procesi 模块和秩变体

我们在 $\mathbb{Q}[x_1,\dots,x_n]$ 中引入了一个理想族 $I_{n,\lambda,s}$ 用于 $\lambda$ 的分区 $k\leq n$ 和一个整数$s \geq \ell(\lambda)$。该族包含作为特例的Haglund-Rhoades-Shimozono的Tanisaki理想$I_\lambda$和理想$I_{n,k}$。我们研究相应的商环 $R_{n,\lambda,s}$ 作为对称群模块。当 $n=k$ 和 $s$ 是任意的时，我们恢复 Garsia-Procesi 模块，当 $\lambda=(1^k)$ 和 $s=k$ 时，我们恢复 Haglund-Rhoades 的广义共变代数-下园。我们给出了$R_{n,\lambda,s}$的单项式基，统一了Garsia-Procesi和Haglund-Rhoades-Shimozono研究的单项式基，实现了$R_{n,\ lambda,s}$ 表示对 $(n,\lambda,s)$ 有序集合分区的操作。我们还证明了 $R_{n,\lambda,s}$ 的希尔伯特级数和分级 Frobenius 特征的公式。然后，我们使用 Weyman 的结果将我们的工作与 Eisenbud-Saltman 等级品种联系起来。作为我们工作的应用，我们给出了秩变体与对角矩阵的方案-理论交的坐标环的单项式基、希尔伯特级数公式和分级Frobenius特征公式。