A crystal rule for GL-multiplicities of quiver loci

A type A quiver representation consists of vector spaces and linear maps arranged along a path. After fixing the dimensions, a product of general linear groups acts by change of basis, and the closures of its orbits are called quiver loci. We present a positive combinatorial rule for the multiplicities of irreducible GL-representations in their coordinate rings, for arbitrary orientations of a type A quiver. The rule is proved by developing a crystal structure on certain nonnegative integer matrices; the multiplicities are counted by highest-weight elements. This talk is based on joint work (arXiv:2608.06664) with Ian Cavey and Andrew Hardt.