Matroid polytopes and their volumes

Abstract

We express the matroid polytope $P_M$ of a matroid $M$ as a signed Minkowski sum of simplices, and obtain a formula for the volume of $P_M$. This gives a combinatorial expression for the degree of an arbitrary torus orbit closure in the Grassmannian $G{r}_{k,n}$. We then derive analogous results for the independent set polytope and the associated flag matroid polytope of $M$. Our proofs are based on a natural extension of Postnikov's theory of generalized permutohedra.

15 pages, 2 figures. This version corrects a small error in the definition of generalized permutahedra and of Q-polytopes. All the theorems hold without modification