Mario Sanchez: Valuations and the Hopf monoid of Generalized Permutahedra

Abstract

Many combinatorial objects, such as matroids, graphs, and posets, can be realized as polytopes - specifically, as generalized permutahedra. This realization respects the natural multiplication of these objects as well as natural "breaking" operations. Surprisingly many of the important invariants of these objects, when viewed as functions on polytopes, satisfy an inclusion-exclusion formula with respect to subdivisions. Functions which satisfy this formula are known as valuations. In this talk, I will discuss recent work with Federico Ardila that completely describes the relationship between the algebraic structure on generalized permutahedra and valuations. Our main contribution is a new easy-to-apply method that converts simple valuations into more complicated ones. This method unifies the majority of valuations on matroid polytopes as well as giving new valuations on posets and building sets.