Valuative invariants for large classes of matroids

Abstract: Valuations on polytopes are maps that combine the geometry of polytopes with relations in a group. It turns out that many important invariants of matroids are valuative on the collection of matroid base polytopes, e.g., the Tutte polynomial and its specializations or the Hilbert–Poincaré series of the Chow ring of a matroid.

In this talk I will present a framework that allows us to compute such invariants on large classes of matroids, e.g., (sparse) paving and elementary split matroids, explicitly. The concept of split matroids introduced by Joswig and myself is relatively new and generalize the notion of (sparse) paving matroids. These classes appear naturally in the context of valuations and proved to be useful in other cases, too. other cases. I will demonstrate our framework by looking at Ehrhart polynomials and further examples.

This talk is based on the preprint `Valuative invariants for large classes of matroids'. On special request I will also mention the main result of `The Merino-Welsh conjecture for split matroids' which discusses a well known conjecture on values of the Tutte polynomial. Both of these articles are joint work with Luis Ferroni.

computational geometrydiscrete mathematicscommutative algebracombinatorics

Audience: researchers in the topic

Copenhagen-Jerusalem Combinatorics Seminar

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