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

Abstract: Coinvariant rings $R_n$ are a well-studied family of rings with rich connections to the combinatorics of the symmetric group $S_n$. Two remarkable families of graded rings which generalize the coinvariant rings are:

• The cohomology rings of Springer fibers $R_\lambda$, whose $S_n$-module structure coincides with the dual Hall-Littlewood functions under the graded Frobenius characteristic map.

• The generalized coinvariant rings $R_{n,k}$ of Haglund, Rhoades, and Shimozono, which give a representation-theoretic interpretation of the expression in the Delta Conjecture when $t=0$.

In this talk, we introduce a family of graded rings $R_{n,\lambda,s}$ which are a common generalization of $R_\lambda$ and $R_{n,k}$. We then generalize many of the previously known formulas for $R_{\lambda}$ and $R_{n,k}$ to our setting. Finally, we show how our results can be applied to Eisenbud-Saltman rank varieties, generalizing work of De Concini-Procesi and Tanisaki.

combinatoricsmetric geometry

Audience: researchers in the topic

Comments: There is a pre-seminar (aimed at graduate students) at 3:30–4:00 PM (US Pacific time, UTC -7). The main talk starts at 4:10.

UW combinatorics and geometry seminar