Tropical Incidence Relations, Polytopes, and Concordant Matroids

Abstract

In this paper, we develop a tropical analog of the classical flag variety that we call the flag Dressian. We find relations, which we call "tropical incidence relations", for when one tropical linear space is contained in another, and show that the flag Dressian is a tropical prevariety. In the case of 2-step flag Dressians, which we call "tropical incidence prevarieties", we find an equivalence between points in this space and induced subdivisions of a hypersimplex, generalizing two parts of an equivalence given by D. Speyer for tropical linear spaces. We attempt to generalize the third part of Speyer's equivalence to concordant matroids and obtain some partial results.