The Weak-Map Order and Polytopal Decompositions of Matroid Base Polytopes

TL;DR

This paper explores the relationship between the weak-map order and polytopal decompositions of matroid base polytopes, classifies matroids into five types based on these properties, and provides counterexamples to a conjecture about their structure.

Contribution

It classifies matroids into five types regarding weak-map order and decomposability, and presents counterexamples to Lucas's conjecture.

Findings

Matroids can be classified into five types based on weak-map order and decomposability.

Counterexamples show Lucas's conjecture does not hold in general.

The study clarifies the relation between matroid base polytope decompositions and the weak-map order.

Abstract

The weak-map order on the matroid base polytopes is the partial order defined by inclusion. Lucas proved that the base polytope of no binary matroid includes the base polytope of a connected matroid. A matroid base polytope is said to be decomposable when it has a polytopal decomposition which consists of at least two matroid base polytopes. We shed light on the relation between the decomposability and the weak-map order of matroid base polytopes. We classify matroids into five types with respect to the weak-map order and decomposability. We give an example of a matroid in each class. Moreover, we give a counterexample to a conjecture proposed by Lucas, which says that, when one matroid base polytope covers another matroid base polytope with respect to inclusion, the latter matroid base polytope should be a facet of the former matroid base polytope.