Seminar on Combinatorics, Lie Theory, ​and Topology

A raising operator formula for Macdonald polynomials

by George H. Seelinger (University of Michigan)

Aula Riunioni (Department of Mathematics)

Aula Riunioni

Department of Mathematics


Macdonald polynomials are a basis of symmetric functions with coefficients in $\mathbb{Q}(q,t)$ exhibiting deep connections to representation theory and algebraic geometry. In particular, specific specializations of the $q,t$ parameters recover various widely studied bases of symmetric functions, such as Hall-Littlewood polynomials, Jack polynomials, q-Whittaker functions, and Schur functions. Central to this study is the fact that the Schur function basis expansion of the Macdonald polynomials have coefficients which are polynomials in $q,t$ with nonnegative integer coefficients, which can be realized via a representation-theoretic model. A more combinatorial approach to this result lies in first expanding Macdonald polynomials into LLT polynomials via the work of Haglund-Haiman-Loehr. LLT polynomials were first introduced by Lascoux-Leclerc-Thibbon as a q-deformation of a product of Schur polynomials and have subsequently appeared in the study of Macdonald polynomials and related families. In this talk, I will explain this background and provide a new explicit "raising operator" formula for Macdonald polynomials that follows from a realization of LLT polynomials in the elliptic Hall algebra of Burban and Schiffmann, which we describe via an isomorphism between the shuffle algebra studied by Feigin and Tsymbaliuk and part of the elliptic Hall algebra. This work is joint with Jonah Blasiak, Mark Haiman, Jennifer Morse, and Anna Pun.