I describe a new isomorphism invariant of matroids that is a quasisymmetric function. This invariant
This last property leads to an interesting application: it can sometimes be used to prove that a matroid base polytope has no decompositions into smaller matroid base polytopes. From work of Lafforgue, the lack of such a decomposition implies the matroid has only a finite number of vector representations up to projective equivalence.
This is joint work with Ning Jia and Victor Reiner.
The paper is accessible at http://arxiv.org/abs/math.CO/0606646.