Multiline queue crystals and Demazure atom positivity

Speaker: Olya Mandelshtam

Affiliation: University of Waterloo

Abstract

Multiline queues are combinatorial objects originally introduced to study probabilities in interacting particle systems such as the totally asymmetric simple exclusion process (TASEP). More recently, they have also appeared in algebraic combinatorics through formulas for important families of symmetric functions, such as Schur functions and q-Whittaker polynomials. These formulas can be refined into nonsymmetric pieces called Demazure atoms and TASEP polynomials. 

Crystal structures are a standard tool for studying Schur expansions and positivity phenomena in algebraic combinatorics. In this talk, I will describe a crystal structure on multiline queues and explain how it can be used to prove that the TASEP polynomials expand positively in the Demazure atom basis. As a consequence, we obtain new positivity results for a broader family of nonsymmetric q-Whittaker polynomials.

This is joint work with Harper Niergarth and Kartik Singh.