Matroid Polytopes and Their Volumes

Abstract

International audience 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. On exprime le polytope matroïde $P_M$ d'un matroïde $M$ comme somme signée de Minkowski de simplices, et on obtient une formule pour le volume de $P_M$. Ceci donne une expression combinatoire pour le degré d'une clôture d'orbite de tore dans la Grassmannienne $Gr_{k,n}$. Ensuite, on déduit des résultats analogues pour le polytope ensemble indépendant et pour le polytope matroïde drapeau associé à $M$. Nos preuves sont fondées sur une extension naturelle de la théorie de Postnikov de permutoèdres généralisés.

Generalized permutahedra: Minkowski linear functionals and Ehrhart positivity

“…Towards the conjecture we prove in Theorem 4.5 that the linear coefficient for any generalized permutahedron is indeed nonnegative. 1 As an application, we then obtain an inequality among beta invariants of contractions of any given matroid in Corollary 4.7 using a result of Ardila, Benedetti and Doker [3]. We also apply our results to solid-angle polynomials and show the existence of a three dimensional generalized permutahedron whose solid-angle polynomial has negative linear term.…”

“…We say that i y i P i defines a polytope if P = i : y i <0 (−y i )P i is a Minkowski summand of Q = i : y i ≥0 y i P i , in which case i y i P i represents the Minkowski difference Q−P . In [3], Ardila, Benedetti and Doker showed that every generalized permutahedron has a unique expression as a signed Minkowski sum of standard simplices. By a result of Shephard, Minkowski summands of polytopes can be characterized in terms of their edge directions and edge lengths (see [21, p. 318]).…”

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We characterize all signed Minkowski sums that define generalized permutahedra, extending results of Ardila-Benedetti-Doker (2010). We use this characterization to give a complete classification of all positive, translation-invariant, symmetric Minkowski linear functionals on generalized permutahedra.

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“…Generalized permutahedra form a combinatorially rich class of polytopes that naturally appear in many areas of mathematics such as combinatorics, geometry, representation theory, optimization and statistics (see, e.g., [3,11,18,20,24,30,31,33,34]). Introduced by Postnikov [34] as deformations of the permutahedron, they comprise many other significant classes of polytopes, such as matroid polytopes, associahedra and Stanley-Pitman polytopes, and have been shown to be equivalent to M -convex polyhedra in discrete analysis [32] and polymatroids in optimization which have been intensively studied since the 1970s [16,19].…”

On the Ehrhart Polynomial of Minimal Matroids

“…In [7] Castillo and Liu conjectured something stronger regarding Ehrhart positivity: that all generalized permutohedra are Ehrhart positive. The validity of that conjecture implies that all matroids are Ehrhart positive, since it is known that matroid polytopes are a subfamily of generalized permutohedra (see for example [1]).…”