Springer fibers, rank varieties, and generalized coinvariant rings

Abstract: Springer fibers are a family of varieties with the remarkable property that their cohomology rings $R_\lambda$ have the structure of a symmetric group module, even though there is no $S_n$ action on the varieties themselves. This is one of the first examples of a geometric representation. In the 80s, De Concini and Procesi proved that $R_\lambda$ has another geometric description as the coordinate ring of the scheme-theoretic intersection of a nilpotent orbit closure with diagonal matrices. This led them to an explicit presentation for $R_\lambda$ in terms of generators and relations, which was further simplified by Tanisaki. In this talk, we present a generalization of this work to the coordinate ring of a scheme-theoretic intersection of Eisenbud-Saltman rank varieties. We then connect these coordinate rings to the generalized coinvariant rings recently introduced by Haglund, Rhoades, and Shimozono in their work on the Delta Conjecture from Algebraic Combinatorics. We then give combinatorial formulas for the Hilbert series and graded Frobenius series of our coordinate rings generalizing those of Haglund-Rhoades-Shimozono and Garsia-Procesi.

algebraic geometry

Audience: researchers in the topic

UC Davis algebraic geometry seminar