Seminar - Jenna Rajchgot

Title: Type A quiver loci

Abstract:  A quiver is a finite directed graph and a representation of a quiver is an assignment of vector space to each vertex and linear map to each arrow. Once the vector spaces at each vertex have been fixed, the space of representations is an algebraic variety. This variety carries an action of a product of general linear groups, which acts by change of basis. For example, the orbits associated to a quiver with one vertex and a loop are in bijection with Jordan normal forms. I’ll focus on the setting where the quiver’s underlying graph is a type A Dynkin diagram, and discuss results on the geometry and combinatorics of the associated orbit closures (a.k.a. quiver loci). I’ll show that each quiver locus is isomorphic, up to smooth factor, to a patch of a Schubert variety, and explain how orbit closure containment is determined by Bruhat order on the symmetric group. I’ll also describe combinatorial formulas for multidegrees and K-polynomials. This is joint work with Ryan Kinser and Allen Knutson.