Algebraic Geometry and Combinatorics Seminar: Expansions of chromatic quasisymmetric functions via the A_{q,t} algebra

Abstract: The Stanley-Stembridge conjecture asserts that the chromatic symmetric functions of incomparability graphs of (3+1)-free posets are e-positive. In 2024, Hikita proved the conjecture by guessing a remarkable formula for the expansion coefficients and proving they satisfy the modular law recurrence. We give a second proof of Hikita's formula by direct algebraic methods involving the A_{q,t} algebra of Carlsson and Mellit. In fact, we prove a "master formula" for chromatic symmetric functions in terms of Macdonald polynomials that involve an extra parameter t. Upon specializing t to different values, we obtain expansions into various bases of symmetric functions, including Hikita's elementary basis expansion. Along the way, I'll explain what this all has to do with the combinatorics of parabolic flag Hilbert schemes.

This is joint work with Anton Mellit, Marino Romero, Kevin Weigl, and Joshua Jeishing Wen.

Host: Martha Precup