Valuations and Hopf Monoid of Generalized Permutahedra

Abstract: Many combinatorial objects, such as matroids, graphs, and posets, can be realized as generalized permutahedra - a beautiful family of polytopes. These realizations respect 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 are valuations, that is, they satisfy an inclusion-exclusion formula with respect to subdivisions. In this talk, I will discuss work with Federico Ardila that 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. New examples of valuations coming from this method include the Kazhdan-Lustzig polynomials of matroids and the motivic zeta functions of matroids.

commutative algebraalgebraic geometrycombinatorics

Audience: researchers in the topic

Matroids - Combinatorics, Algebra and Geometry Seminar