Discrete Applied Math Seminar by Bruce Sagan: Chromatic Symmetric Functions, P-tableaux, and Jacobi-Trudi Determinants
Speaker: Bruce Sagan, Michigan State University
Title: Chromatic Symmetric Functions, P-tableaux, and Jacobi-Trudi Determinants
Abstract: For a graph G, let X(G) be Stanley’s chromatic symmetric function. The Stanley-Stembridge Conjecture states that if P is a (3+1)-free poset and G(P) is its incomparability graph then X(G(P)) has nonnegative coefficients when expanded in the elementary symmetric function basis. This is one of the most important open problems about X(G) and has connections with Hessenberg varieties in algebraic geometry. We explore an approach to this conjecture which we believe has been overlooked by the combinatorial community and which involves Gasharov’s P-tableaux and the dual Jacobi-Trudi determinants. In particular, we can give a new combinatorial interpretation to the coefficient of e_n in X(G(P)) for any (3+1)-free P with #P = n, and give a simple proof the if the Young diagram of λ has at most two columns then the coefficient of e_λ is nonnegative for any such P. This is joint work with Zachary Hamaker.
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