Combinatorics Seminar: A noncommutative Schur function approach to chromatic symmetric functions

Abstract: The chromatic symmetric function X_G of a graph G was introduced by Richard Stanley in 1995 as a generalization of the chromatic polynomial. The chromatic symmetric function has particularly interesting properties when G is the incomparability graph of a (3+1)-free poset. In this case, X_G has positive integer coefficients when expanded in the basis of Schur functions, and the Stanley-Stembridge conjecture states that X_G has positive integer coefficients when expanded in the basis of elementary symmetric functions. For a (3+1)-free poset P, we define a ring of P-analogues of symmetric functions in noncommuting variables and reformulate the Stanley-Stembridge conjecture into a conjecture about this ring. With this formulation, we show that the coefficient of the elementary symmetric function e_\lambda is a non-negative integer when \lambda is a two-column, hook, or rectangular shape. This talk is based on joint work with Jonah Blasiak, Holden Eriksson, and Pavlo Pylyavskyy.

Host: Martha Precup