Characterizations of certain matroids by maximizing valuative invariants

TL;DR

This paper characterizes certain matroids as vertices of a polytope formed by maximizing valuative invariants, confirming conjectures and providing new examples of such matroids.

Contribution

It proves that specific classes of matroids are vertices of the polytope, confirming conjectures and extending the list of known extremal matroids.

Findings

Vertices include cycle matroids of complete graphs, projective geometries, and Dowling geometries.

Confirmed that direct sums of uniform matroids are vertices.

Extended to direct sums of certain extremal matroids.

Abstract

Luis Ferroni and Alex Fink recently introduced a polytope of all unlabeled matroids of rank $r$ on $n$ elements, and they showed that the vertices of this polytope come from matroids that can be characterized by maximizing a sequence of valuative invariants. We prove that a number of the matroids that they conjectured to yield vertices indeed do (these include cycle matroids of complete graphs, projective geometries, and Dowling geometries), and we give additional examples (including truncations of cycle matroids of complete graphs, Bose-Burton geometries, and binary and free spikes with tips). We prove a special case of a conjecture of Ferroni and Fink by showing that direct sums of uniform matroids yield vertices of their polytope, and we prove a similar result for direct sums whose components are in certain restricted classes of extremal matroids.