A combinatorial rule for $GL$-multiplicities of $A_n$-quiver loci

Abstract

We give the first positive combinatorial rule for the multiplicities of irreducible $GL$-representations in the coordinate rings of type $A$ quiver orbit closures, valid for every orientation of the quiver. Previously, for the special case of varieties of complexes, work of De Concini--Strickland in the early 1980s gave an implicit description of these multiplicities. The combinatorial objects in our rule carry a crystal structure whose highest-weight elements compute these multiplicities.