Matroid polytopes and their volumes

TL;DR

This paper provides a combinatorial formula for the volume of matroid polytopes by expressing them as Minkowski sums of simplices, linking matroid theory with algebraic geometry.

Contribution

It introduces a novel Minkowski sum decomposition for matroid polytopes and derives volume formulas, extending Postnikov's generalized permutohedra theory.

Findings

Volume formula for matroid polytopes

Expression of matroid polytopes as Minkowski sums

Connection to torus orbit degrees in Grassmannians

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 $Gr_{k,n}$. 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.