A combinatorial model for Macdonald polynomials via hopping particles
- Olya Mandelshtam (University of Waterloo)
Over the last couple of decades, the theory of interacting particle systems has found some unexpected connections to orthogonal polynomials, symmetric functions, and various combinatorial structures. The asymmetric simple exclusion process (ASEP) has played a central role in this study: recently, it was discovered that its probabilities can be written as specializations of Macdonald polynomials, which are an important family of symmetric functions with parameters q and t that generalize the Schur functions. In this talk, we describe a combinatorial model called a "multiline queue" that illuminates this remarkable connection. One one hand, multiline queues give a new formula for symmetric Macdonald polynomials, and on the other hand they encode the dynamics of the ASEP and give formulas for its stationary distribution. As an application, we consider a specialization of multiline queues as a natural setting to recover some classical results in symmetric function theory.