Combinatorics Seminar: Combinatorics of Koornwinder polynomials at q=t and exclusion processes

Seminar | December 4 | 12-1 p.m. | 939 Evans Hall

 Sylvie Corteel, MSRI

 Department of Mathematics

I will explain how to build Koornwinder polynomials at q = t from moments of Askey-Wilson polynomials. I will use the combinatorial theory of Viennot for orthogonal polynomials and their moments. An extension of this theory allows to build multivariate orthogonal polynomials. The key step for this construction are a Cauchy identity for Koornwinder polynomials and a Jacobi-Trudi formula for the 9th variation of Schur functions. This gives us an elegant path model for these polynomials. I will also explain a positivity conjecture for these polynomials that we can prove in several special cases. For this, we link them to the stationary distribution of an exclusion process and prove positivity by exhibiting a combinatorial model called rhombic staircase tableau. This talk is based ongoing work with Olya Mandelshtam (Brown) and Lauren Williams (Berkeley). Several open problems for graduate students will be proposed as I will be in Berkeley during the Spring and the Summer 2018.