Abstract:

The LLT polynomials introduced by Lascoux, Leclerc, and Thibon are a family of symmetric polynomials indexed by tuples of partitions that generalize the more well-known Schur polynomials. In this talk will show how to construct a version of theses LLT polynomials as the partition function of a Yang-Baxter integrable vertex model. As a consequence, we will be able to prove that the polynomials are in fact symmetric and that they satisfy a certain Cauchy identity. Building on this we will define coupled domino tilings of the Aztec diamond and use our vertex model to enumerate them. Finally, if time permits, we will present an algorithm for determining when the LLT polynomial indexed by the pair of partitions $(\lambda^{(1)},\lambda^{(2)})$ is equal to the LLT polynomial indexed by $(\lambda^{(2)},\lambda^{(1)})$. The key ingredient will be the vertex model construction of the polynomials.