Gelfand–Tsetlin crystals

Abstract

Abstract We give a crystal structure on the set of Gelfand–Tsetlin patterns (GTPs), which parametrize bases for finite-dimensional irreducible representations of the general linear Lie algebra. The crystal data are given in closed form and are expressed using tropical polynomial functions of the entries of the patterns. We prove that with this crystal structure, the natural bijection between GTPs and semistandard Young tableaux is a crystal isomorphism.