  We use the equivariant cohomology ring of the permutohedral variety to study
matroids and their invariants. Investigating the pushforward of matroid Chern
classes defined by A. Berget, C. Eur, H. Spink and D. Tseng to the product
space $\mathbb{P}^n \times \mathbb{P}^n$, we establish an equivariant
generalization of the Tutte polynomial of a matroid. This was suggested in a
survey paper by M.Micha{\l}ek. We discuss how this polynomial encodes
properties of the matroid by looking at special evaluations. We further
introduce an equivariant generalization of the reduced characteristic
polynomial of a matroid.
