Harvard-MIT Combinatorics Seminar: Geometry of the Chromatic Symmetric Function of Trees


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March 8, 2023 4:15 pm - 5:15 pm
Science Center 232

Mario Sanchez - Cornell

Stanley's chromatic symmetric function is a generalization of the chromatic polynomial of a graph that encodes coloring information for graphs. One open conjecture is that non-isomorphic trees have different chromatic symmetric functions. In this talk, I will give two geometric interpretations of these functions for trees. The first interprets the chromatic symmetric function of a tree as an element in the Chow ring of the permutahedral variety opening the conjecture to algebraic geometric methods. The second describes this open conjecture in terms of the theory of valuations on generalized permutahedra. These are functions on polytopes which satisfy certain inclusion-exclusion relations with respect to subdivisions. From this perspective, we make progress on the conjecture by constructing new valuations on generalized permutahedra. We will primarily focus on this convex geometric interpretation for this talk.