Sean Griffin's Pre-Seminar, May 27, 2020

Sean Griffin's Pre-Seminar 2

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

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

The cohomology rings of Springer fibers Rλ, whose Sn-module structure coincides with the dual Hall-Littlewood functions under the graded Frobenius characteristic map.
The generalized coinvariant rings Rn,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 Rn,λ,s which are a common generalization of Rλ and Rn,k. We then generalize many of the previously known formulas for Rλ and Rn,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.