Schur and LLT Polynomials from Lattice Models

In this talk, we will use lattice models from statistical physics to study symmetric polynomials (in particular, Schur and LLT polynomials). Using the bijection between semi-standard Young tableaux and non-intersecting lattice path, we give a 5-vertex lattice model interpretation for Schur polynomials, and generalize this construction to LLT polynomials which are generating functions of ribbon Tableaux. I will show that our lattice models are solvable, meaning that they satisfy a Yang-Baxter equation, which implies that the polynomials are indeed symmetric. If time permits, I will also discuss how the Yang-Baxter equation can be used to prove identities of symmetric polynomials.

Sylvester Zhang (University of Minnesota)