The polytope of all matroids in ranks 2 and 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 $n\le 33$ and ${\Omega }_{3,n}$ for $n\le 10$. Further, we compute Schubert expansions for all isomorphism classes of matroids of rank $2$ up to $n=80$, and for rank $3$ up to $n=11$.

23 pages, 3 figures