Seminar

# Compact formulas for modified Macdonald polynomials

#### Algebraic and Enumerative Combinatorics

#### 30 January 14:30 - 15:20

#### Olya Mandelshtam - Brown University

The asymmetric simple exclusion exclusion process (ASEP) is a one-dimensional model of hopping particles that has been extensively studied in statistical mechanics, probability, and combinatorics. It also has remarkable connections with orthogonal symmetric polynomials in many variables such as Macdonald and Koornwinder polynomials. In this talk we will give an overview of some recent work, and discuss how we can use multiline queues, a new object recently introduced by James Martin to compute stationary probabilities of the ASEP on a circle, to obtain new combinatorial formulas for both the symmetric Macdonald polynomials $P_{\lambda}(X;q,t)$ and the nonsymmetric Macdonald polynomials $E_{\lambda}(X;q,t)$, where $\lambda$ is a partition. Building on this result, we obtain new, compact formulas for modified Macdonald polynomials $H_{\lambda}(X;q,t)$. Finally, we introduce a new quasisymmetric analogue of Macdonald polynomials, which refine the $P_{\lambda}$'s and specialize at $q=t=0$ to the quasisymmetric Schur polynomials defined by Haglund, Luoto, Mason, and van Willigenburg.

Organizers

Sara Billey

University of Washington

Petter Brändén

KTH Royal Institute of Technology

Sylvie Corteel

Université Paris Diderot, Paris 7

Svante Linusson

KTH Royal Institute of Technology