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

          @misc{ pirsa_PIRSA:20060030,
            doi = {10.48660/20060030},
            url = {https://pirsa.org/20060030},
            author = {Rajchgot, Jenna},
            keywords = {Mathematical physics},
            language = {en},
            title = {Type D quiver representation varieties, double Grassmannians, and symmetric varieties},
            publisher = {Perimeter Institute},
            year = {2020},
            month = {jun},
            note = {PIRSA:20060030 see, \url{https://pirsa.org}}
          }
          

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) x 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.