Matroid polytopes and their volumes

Federico Ardila; Carolina Benedetti; Jeffrey Doker

Matroid polytopes and their volumes

Combinatorics 2013-01-17 v3 Algebraic Geometry

Authors: Federico Ardila , Carolina Benedetti , Jeffrey Doker

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.

Keywords

intrinsic volumes of polytopes matroid theory polytopes

Cite

@article{arxiv.0810.3947,
  title  = {Matroid polytopes and their volumes},
  author = {Federico Ardila and Carolina Benedetti and Jeffrey Doker},
  journal= {arXiv preprint arXiv:0810.3947},
  year   = {2013}
}

Comments

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

Matroid polytopes and their volumes

Abstract

Keywords

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