Type D quiver representation varieties, double Grassmannians, and symmetric varieties

Abstract: Since the 1980s, mathematicians have found connections between orbit closures in type $A$ quiver representation varieties and Schubert varieties in type $A$ flag varieties. For example, singularity types appearing in type $A$ quiver orbit closures coincide with those appearing in Schubert varieties in type $A$ flag varieties (Bobinski-Zwara); combinatorics of type $A$ quiver orbit closure containment is governed by Bruhat order on the symmetric group (follows from work of Zelevinsky, Kinser-R.); and multiple researchers have produced formulas for classes of type $A$ quiver orbit closures in equivariant cohomology and $K$-theory in terms of Schubert polynomials, Grothendieck polynomials, and related objects. After recalling some of this type $A$ story, I will discuss joint work with Ryan Kinser on type $D$ quiver representation varieties. I will describe explicit embeddings which completes a circle of links between orbit closures in type $D$ quiver representation varieties, $B$-orbit closures (for a Borel subgroup $B$ of $GL_n$) in certain symmetric varieties $GL_n/K$, and $B$-orbit closures in double Grassmannians $Gr(a, n) \times Gr(b, n)$. I will end with some geometric and combinatorial consequences, as well as a brief discussion of joint work in progress with Zachary Hamaker and Ryan Kinser on formulas for classes of type $D$ quiver orbit closures in equivariant cohomology.

mathematical physicsalgebraic geometryrepresentation theory

Audience: researchers in the topic

Geometric Representation Theory conference

Series comments: Originally planned as a twinned conference held simultaneously at the Max Planck Institute in Bonn, Germany and the Perimeter Institute in Waterloo, Canada. The concept was motivated by the desire to reduce the environmental impact of conference travels. In order to view the talks, register at the website: www.mpim-bonn.mpg.de/grt2020 . The talks from previous days can be be viewed at pirsa.org/C20030 ; slides from the talks are posted here: www.dropbox.com/sh/cjzqbqn7ql8zcjv/AAANB82Hh4t5XDc5RPcZzW0Aa?dl=0