Tropical vector bundles and matroids

We introduce a notion of tropical vector bundle on a tropical toric variety which is a tropical analogue of a torus equivariant vector bundle on a toric variety. Alternatively it can be called a toric matroid bundle. We define equivariant $K$-theory and characteristic classes of these bundles. As a particular case, we show that any matroid comes with tautological tropical toric vector bundles over the permutahedral toric variety and the corresponding equivariant $K$-classes and Chern classes recover the tautological classes of matroids constructed in the recent work of Berger-Eur-Spink-Tseng. In analogy with toric vector bundles, we define sheaf of sections and Euler characteristic as well as positivity notions such as global generation, ampleness and nefness for tropical toric vector bundles. Moreover, we prove a vanishing of higher cohomologies result. Finally, we study the splitting of our tropical toric vector bundles and, in particular, an analogue of Grothendieck's theorem on splitting of vector bundles on projective line.