Valuative invariants for large classes of matroids

Luis Ferroni, Benjamin Schröter

Published 2022-08-09Version 1

We study an operation in matroid theory that allows one to transition a given matroid into another with more bases via relaxing a "stressed subset". This framework provides a new combinatorial characterization of the class of elementary split matroids, which is expected to be asymptotically predominant. Moreover, it permits to describe an explicit matroid subdivision of a hypersimplex, which in turn can be used to write down concrete formulas for the evaluations of any valuative invariant on these matroids. This shows that evaluations depend solely on the behavior of the invariant on a well-behaved small subclass of Schubert matroids that we call "cuspidal matroids". Along the way, we make an extensive summary of the tools and methods one might use to prove that an invariant is valuative, and we use them to provide new proofs of the valuativeness of several invariants. We address systematically the consequences of our approach for a comprehensive list of invariants. They include the volume and Ehrhart polynomial of base polytopes, the Tutte polynomial, Kazhdan-Lusztig polynomials, the Whitney numbers of the first and second kind, spectrum polynomials and a generalization of these by Denham, chain polynomials and Speyer's $g$-polynomials, as well as Chow rings of matroids and their Hilbert-Poincar\'e series. The flexibility of this setting allows us to give a unified explanation for several recent results regarding the listed invariants; furthermore, we emphasize it as a powerful computational tool to produce explicit data and concrete examples.