Geodesic
An adjacency criterion for Coxeter matroids
Journal of Algebraic Combinatorics, Tome 9 (1999) no. 3, pp. 271-280.

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Summary: A Coxeter matroid is a generalization of matroid, ordinary matroid being the case corresponding to the family of Coxeter groups $A _{ n }$, which are isomorphic to the symmetric groups. A basic result in the subject is a geometric characterization of Coxeter matroid in terms of the matroid polytope, a result first stated by Gelfand and Serganova. This paper concerns properties of the matroid polytope. In particular, a criterion is given for adjacency of vertices in the matroid polytope.
Keywords: matroid, Coxeter matroid, Coxeter group, matroid polytope 
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     title = {An adjacency criterion for {Coxeter} matroids},
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Borovik, Alexandre; Vince, Andrew. An adjacency criterion for Coxeter matroids. Journal of Algebraic Combinatorics, Tome 9 (1999) no. 3, pp. 271-280. http://geodesic.mathdoc.fr/item/JAC_1999__9_3_a2/