Flagged LLT polynomials and nonsymmetric Macdonald polynomials

LLT polynomials are a family of symmetric functions that serve as a q-deformation of a product of (skew) Schur functions. They can be defined using tableaux combinatorics and have a surprising way of showing up in many positivity problems in symmetric function theory with connections to representation theory and algebraic geometry. Some notable examples include the shuffle theorem, the Haglund-Haiman-Loehr formula for modified Macdonald polynomials, and a relationship with chromatic symmetric functions associated unit interval orders. Recently, in joint work with Blasiak, Haiman, Morse, and Pun, we develop the theory of a nonsymmetric analogue of LLT polynomials we call flagged LLT polynomials, which can be described combinatorially in terms of certain flagged tableaux. We will survey a few of their nice combinatorial and algebraic properties and discuss some applications to other nonsymmetric generalizations of results surrounding the theory of Macdonald polynomials.

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Date published: Monday, April 20, 2026