Ordered set partitions, Garsia-Procesi modules, and rank varieties

Abstract

We introduce a family of ideals $I_{n,\lambda ,s}$ in $\mathbb{Q}[x_1,\dots ,x_n]$ for $\lambda$ a partition of $k\le n$ and an integer $s\ge \ell (\lambda )$. 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.

arXiv comment51 pages. Example 3.15, Figure 6, and Example 4.11 have been added for clarity. Full version of extended abstract in the proceedings of FPSAC 2020