Ordered set partitions, Garsia-Procesi modules, and rank varieties
Ordered set partitions, Garsia-Procesi modules, and rank varieties
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有序集分区、Garsia-Procesi 模块和等级变体
DOI：
10.1090/tran/8237
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发表时间：
2020
期刊：
影响因子：
--
通讯作者：
Sean T. Griffin
中科院分区：
文献类型：
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作者：
Sean T. Griffin
我们在$\mathbb{Q}[x_1,\dots,x_n]$中引入一个理想族$I_{n,\lambda,s}$，对于$\lambda$，一个分区$k\leq n$和一个整数$s \geq \ell(\lambda)$。这个家族既包含谷崎理想$I_\lambda$，也包含作为特例的哈格伦德-罗兹-下野理想$I_{n,k}$。我们研究了相应的商环$R_{n,\lambda,s}$作为对称群模。当$n=k$和$s$为任意时，我们恢复了Garsia-Procesi模，当$\lambda=(1^k)$和$s=k$时，我们恢复了Haglund-Rhoades-Shimozono的广义协不变代数。我们统一了Garsia-Procesi和Haglund-Rhoades-Shimozono研究的单项基，给出了$R_{n,\lambda,s}$的一个单项基，并通过对$(n,\lambda,s)$ -有序集分区的作用实现了$R_{n,\lambda,s}$的$S_n$ -模块结构。我们还证明了$R_{n,\lambda,s}$的Hilbert级数和梯度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秩变体联系起来。作为我们工作的一个应用，我们给出了秩变与对角矩阵的方案论交的坐标环的单项式基、Hilbert级数公式和梯度Frobenius特征公式。
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.
DOI：
10.37236/9610
发表时间：
2021
期刊：
2021
影响因子：
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作者：
Rhoades, Brendon;Yu, Tianyi;Zhao, Zehong
通讯作者：
Zhao, Zehong