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matroid polytopes and their volumes. federico ardila
 

Summary: matroid polytopes and their volumes.
federico ardila
carolina benedetti
jeffrey doker
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 inde-
pendent 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.
1 introduction.
The theory of matroids can be approached from many different points of view; a
matroid can be defined as a simplicial complex of independent sets, a lattice of
flats, a closure relation, etc. A relatively new point of view is the study of matroid
polytopes, which in some sense are the natural combinatorial incarnations of
matroids in algebraic geometry and optimization. Our paper is a contribution in
this direction.
We begin with the observation that matroid polytopes are members of the

  

Source: Ardila, Federico - Department of Mathematics, San Francisco State University

 

Collections: Mathematics