Integrable systems and crystals for edge labeled tableaux

TL;DR

This paper introduces edge Schur functions as generating series over edge labeled tableaux, demonstrating their symmetry, deriving identities, and establishing a crystal structure that connects to combinatorial algorithms.

Contribution

It defines edge Schur functions, formulates them as lattice model partition functions, and develops a crystal structure linking to combinatorial algorithms.

Findings

Edge Schur functions are symmetric polynomials.

Derived a Cauchy-type identity with factorial Schur polynomials.

Established a crystal structure on edge labeled tableaux.

Abstract

We introduce the edge Schur functions $E^{\lambda}$ that are defined as a generating series over edge labeled tableaux. We formulate $E^{\lambda}$ 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 $E^{\lambda}$ and show it intertwines with an uncrowding algorithm.