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 $k\leq n$ 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 $R_{n,\lambda , s}$ in terms of $(n,\lambda , s)$-staircases, unifying the monomial bases studied by Garsia-Procesi and Haglund-Rhoades-Shimozono. We realize the $S_n$-module structure of $R_{n,\lambda , s}$ in terms of an action on $(n,\lambda , s)$-ordered set partitions. We find a formula for the Hilbert series of $R_{n,\lambda , s}$ in terms of inversion and diagonal inversion statistics on a set of fillings in bijection with $(n,\lambda , s)$-ordered set partitions. Furthermore, we prove an expansion of the graded Frobenius characteristic of our rings into Gessel’s fundamental quasisymmetric basis.

We connect our work with Eisenbud-Saltman rank varieties using results of Weyman. As an application of our results on $R_{n,\lambda , s}$, 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.