Flagged LLT polynomials and their applications

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 how flagged LLTs can be used to give nonsymmetric generalizations of certain results stated with symmetric functions.

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Date published: Monday, July 6, 2026