Matroid products in tropical geometry

Symmetric powers of matroids were first introduced by Lovasz (Combinatorial surveys, in: Proceedings 6th British combinatorial conference, pp 45-86, 1977) and Mason (Algebr Methods Graph Theory 1:519-561, 1981) 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.