An invitation of the q-Whittaker polynomials, talk 1

Abstract: The q-Whittaker polynomials are a family of symmetric functions that can be obtained as a degeneration of the famous Macdonald polynomials. They have played an important role in integrable probability, notably via the framework of Macdonald processes and their connection to the q-TASEP.

The aim of these lectures will be to give two combinatorial formulas for the q-Whittaker polynomials, using the theory of integrable vertex models. These formulas look completely different, but both of them exhibit the q-positivity of the q-Whittaker polynomials in an explicit way. In reaching this goal, we will pass through a number of important landmarks in the theory, including the coloured stochastic six-vertex model, fusion, and the Yang–Baxter equation.