Valuations for matroid polytope subdivisions

TL;DR

This paper establishes that the combinatorial data of ranks and activities in a matroid can be used to define valuations on subdivisions of matroid polytopes, linking matroid theory with polyhedral valuations.

Contribution

It introduces a novel connection between matroid invariants and valuations on polytope subdivisions, expanding the understanding of matroid polytope structure.

Findings

Ranks and activities define valuations for matroid polytope subdivisions

Valuations are determined by subset ranks and basis activities

Provides a new tool for analyzing matroid polytope decompositions

Abstract

We prove that the ranks of the subsets and the activities of the bases of a matroid define valuations for the subdivisions of a matroid polytope into smaller matroid polytopes.