The polytope of all matroids

TL;DR

This paper explores the polytope formed by all matroids, representing them as lattice points in a high-dimensional space, and investigates the structure and extremal properties of this polytope.

Contribution

It introduces a novel polytope framework for matroids, characterizes its vertices and faces, and links matroid classes to geometric features of the polytope.

Findings

Famous matroid classes appear as faces of the polytope

Explicit dimensions of certain faces are determined

Existence of valuative invariants with specific positivity properties

Abstract

It is possible to write the indicator function of any matroid polytope as an integer combination of indicator functions of Schubert matroid polytopes. In this way, every matroid on $n$ elements of rank $r$ can be thought of as a lattice point in the space having a coordinate for each Schubert matroid on $n$ elements of rank $r$. We study the convex hull of all these lattice points, with particular focus on the vertices, which come from the matroids we call extremal matroids. We show that several famous classes of matroids arise as faces of the polytopes, and in many cases we determine the dimension of this face explicitly. As an application, we show that there exist valuative invariants that attain non-negative values at all representable matroids, but fail to be non-negative in general.