Combinatorics & algebra seminar-GL-multiplicities of type A quiver loci

Title: GL-multiplicities of type A quiver loci
Speaker: Andy Hardt, Towson
Date and time: Monday, October 5, 4–5 pm
Place: Rome 459

Abstract: A product of general linear groups acts on the space of representations of a quiver by change of basis. When the quiver is of type A, the orbit closures are described by rank conditions. These varieties generalize classical degeneracy loci. Their equations are understood through work of Zelevinsky and Kinser–Rajchgot, and their cohomology classes and K-polynomials have been computed by many authors, including Buch–Fulton, Knutson–Miller–Shimozono, Buch–Rimányi and Kinser–Knutson–Rajchgot.

The coordinate ring of a quiver orbit closure is therefore a rational representation of this product of general linear groups. In the special case of varieties of complexes, its irreducible decomposition is implicit in work of De Concini and Strickland. In joint work with Ian Cavey and Alexander Yong, we give a positive combinatorial rule for this decomposition, valid for all type A orbit closures in every orientation. Our rule counts nonnegative integer matrices satisfying admissibility conditions that correspond to the rank conditions of the orbit closure. These matrices carry a crystal structure, and the irreducible multiplicities are counted by the highest-weight admissible matrices.