Matroid polytopes and their volumes

Abstract

We express the matroid polytope PM of a matroid M as a signed Minkowski sum of simplices, and obtain a formula for the volume of PM. This gives a combinatorial expression for the degree of an arbitrary torus orbit closure in the Grassmannian Grk,n. We then derive analogous results for the independent set polytope and the underlying flag matroid polytope of M. Our proofs are based on a natural extension of Postnikov's theory of generalized permutohedra. © 2009 Springer Science+Business Media, LLC.