Georgia Institute of Technology College of Sciences

A toric vector bundle is a vector bundle over a toric variety equipped with a linear action by the torus of the base. Toric vector bundles pf rank r were famously classified by Klyachko (1989) using certain combinatorial data of compatible filtrations in an r-dimensional vector space E. This data can be thought of as a higher rank generalization of an (integer-valued) piecewise linear function. In this talk, we give an interpretation of Klyachko data as a "piecewise linear map" to a tropical linear space. This point of view leads us to introduce the notion of a "matroidal vector bundle", a generalization of toric vector bundles to (possibly non-representable) matroids. As a special case and by-product of this construction, one recovers the tautological classes of matroids introduced by Berget, Eur, Spink and Tseng. This is a work in progress with Chris Manon (Kentucky).