Matroid polytopes and their volumes. (English) Zbl 1204.52016

The matroid polytope \(P_{M}\) of a matroid \(M\) is the polytope whose vertices are the characteristic vectors of the bases of the matroid. Matroid polytopes are in a sense the natural incarnations of matroids in algebraic geometry and optimization.
The paper begins with observing that matroid polytopes are generalized permutohedra, as defined by A. Postnikov [Int. Math. Res. Not. 2009, No. 6, 1026–1106 (2009; Zbl 1162.52007)].
Using a natural extension of Postnikov’s theory of generalized permutohedra, the authors 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\(_{k,n }\). The authors then derive analogous results for the independent set polytope and the underlying flag matroid polytope of \(M\).


52B40 Matroids in convex geometry (realizations in the context of convex polytopes, convexity in combinatorial structures, etc.)


Zbl 1162.52007
Full Text: DOI arXiv


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