Integrable systems and crystals for edge labeled tableaux

Abstract

We introduce the edge Schur functions E λ that are defined as a generating series over edge labeled tableaux. We formulate E λ as the partition function for a solvable lattice model, which we use to show they are symmetric polynomials and derive a Cauchy-type identity with factorial Schur polynomials. Finally, we give a crystal structure on edge labeled tableau to give a positive Schur polynomial expansion of E λ and show it intertwines with an uncrowding algorithm.