Thematic Seminar: Combinatorics of multivariate orthogonal polynomials

Seminar | January 25 | 4:10-5 p.m. | 740 Evans Hall

 Sylvie Corteel, CNRS Universite Paris Diderot

 Department of Mathematics

The classical $q$-hypergeometric orthogonal polynomials are assembled into a hierarchy called the $q$-Askey scheme. It is now a classical subject to study the combinatorics of their coefficients and their moments. The polynomials admit a generalization leading to remarkable orthogonal polynomials in several variables. The most general family is the Macdonald-Koornwinder polynomials and Macdonald polynomials associated to any classical root system can be expressed as limits or special cases of Macdonald-Koornwinder polynomials. Understanding the combinatorics of these polynomials is an important open problem. In this talk we will show some recent progress related to special cases of these polynomials.

We will highlight combinatorial formulas for

1. Certain Macdonald-Koornwinder polynomials using exclusion processes with open boundaries and tableaux combinatorics (arXiv:1510.05023)

2. Macdonald polynomials of type A using exclusion processes and multiline queues (arXiv:1811.01024)

3. Multivariate $q$-Little Jacobi polynomials thanks to Lecture Hall Tableaux (arXiv:1804.02489)

This talk will be about enumerative, algebraic and asymptotics combinatorics. No prior knowledge is required. Open problems will be presented.