Raising operator formulas for Macdonald polynomials and other related families, George Seelinger (University of Michigan) -- February 20th, 2025

Macdonald polynomials are a basis of symmetric functions with coefficients in $\mathbb{Q}(q,t)$ exhibiting rich combinatorics with 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, but for which no general combinatorial formula is known. Surrounding this line of inquiry, a rich theory of combinatorics emerged, encoded in symmetric functions with expansions of the Macdonald polynomials into LLT polynomials via the work of Haglund-Haiman-Loehr, among others. LLT polynomials were first introduced by Lascoux-Leclerc-Thibbon as a q-deformation of a product of Schur polynomials and have appeared in various related contexts since their introduction. In this talk, I will explain this background and provide a new explicit "raising operator" formula for Macdonald polynomials, proved via an LLT expansion, with a detour showing how similar raising operator formulas can provide a bridge between algebraic and combinatorial formulations of some other symmetric functions. This work is joint with Jonah Blasiak, Mark Haiman, Jennifer Morse, and Anna Pun.

The seminar co-organizers are Ian Cavey, Andrew Hardt, David Keating, and Alexander Yong.