The polytope of all matroids in ranks 2 and 3

TL;DR

This paper provides explicit recursive constructions and an implementation for the polytope of all matroids in ranks 2 and 3, enabling computations for various ground set sizes and classifications.

Contribution

It introduces a recursive construction method for the matroid polytope in ranks 2 and 3 and provides an implementation for extensive computational analysis.

Findings

Computed $ ext{Omega}_{2,n}$ for n ≤ 33 and $ ext{Omega}_{3,n}$ for n ≤ 10.

Calculated Schubert expansions for all matroid classes up to n=80 (rank 2) and n=11 (rank 3).

Abstract

We give explicit recursive constructions for the polytope of all matroids $\Omega_{r,n}$ in ranks 2 and 3 for all ground set sizes. This polytope was introduced in recent work by Ferroni and Fink as a tool for checking positivity conjectures for valuative invariants. We supplement our theoretical construction by an implementation, which allows for the computation of $\Omega_{2,n}$ for $n\leq 33$ and $\Omega_{3,n}$ for $n\leq 10$. Further, we compute Schubert expansions for all isomorphism classes of matroids of rank $2$ up to $n = 80$, and for rank $3$ up to $n = 11$.