Matroid Products in Tropical Geometry

TL;DR

This paper explores the relationship between matroid symmetric powers and tropical linear spaces, revealing new geometric and algebraic insights, including connectivity and minor-closed properties of certain matroid classes.

Contribution

It establishes an equivalence between valuated matroids with large symmetric powers and tropical linear spaces, and proves properties about matroids with second symmetric power.

Findings

All tropical linear spaces are connected through codimension one.

Matroids with second symmetric power form a minor-closed class.

There are infinitely many forbidden minors for matroids with second symmetric power.

Abstract

Symmetric powers of matroids were first introduced by Lovasz and Mason in the 1970s, where it was shown that not all matroids admit higher symmetric powers. Since these initial findings, the study of matroid symmetric powers has remained largely unexplored. In this paper, we establish an equivalence between valuated matroids with arbitrarily large symmetric powers and tropical linear spaces that appear as the variety of a tropical ideal. In establishing this equivalence, we additionally show that all tropical linear spaces are connected through codimension one. These results provide additional geometric and algebraic connections to the study of matroid symmetric powers, which we leverage to prove that the class of matroids with second symmetric power is minor closed and has infinitely many forbidden minors.