Matroid Products in Tropical Geometry
Nicholas Anderson
TL;DR
The paper establishes a precise equivalence between valuated matroids that admit arbitrarily large symmetric powers and tropical linear spaces that occur as the variety of a tropical ideal, connecting matroid symmetry with tropical geometry. It shows tropical linear spaces are connected through codimension one, enabling a robust link between combinatorial matroid structure and tropical algebraic geometry. The authors prove that the class of matroids with a second symmetric power is minor-closed and has infinitely many forbidden minors, illustrated via the Vámos matroid and Lindström-type families, highlighting intrinsic obstructions to symmetric powers. The work also develops a tropical-geometric toolkit, including tropical ideals, the Dressian, and a Macaulay-matrix construction, to study symmetric powers, realizability, and related algebraic properties of matroids.
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.