In this talk, we will describe how to apply tools from the study of integrable vertex models to the study of symmetric polynomials and tilings. In particular, we will show how one can construct Schur polynomial as partition functions of certain five vertex models. We will use this construction to prove some basic properties of the Schur polynomials such as symmetry and a Cauchy identity. Following this we will show how these same vertex models can be used to enumerate domino tilings of the Aztec diamond. Our main tool will be the celebrated Yang-Baxter equation.
Time permitting, we will introduce a more complicated, but still integrable, vertex model and briefly describe its relationship with LLT polynomials and recently introduced coupled tilings of the Aztec diamond.