Alcove Paths and Gelfand–Tsetlin Patterns

In their study of the equivariant K-theory of the generalized flag varieties G / P , where G is a complex semisimple Lie group, and P is a parabolic subgroup of G , Lenart and Postnikov introduced a combinatorial tool, called the alcove path model. It provides a model for the highest weight crystals with dominant integral highest weights, generalizing the model by semistandard Young tableaux. In this paper, we prove a simple and explicit formula describing the crystal isomorphism between the alcove path model and the Gelfand–Tsetlin pattern model for type A .