Annals of Combinatorics 1 (1997) 123-134

Coxeter Matroid Polytopes

Alexandre V. Borovik, Israel M. Gelfand, and Neil White

Department of Mathematics, UMIST, P.O. Box 88, Manchester M60 1QD, UK

Department of Mathematics, Rutgers University, New Brunswick, NJ 08903, USA

Department of Mathematics, University of Florida, Gainesville, FL 32611, USA

Recieved February 7, 1997

AMS Subject Classification: 05B35, 05E15, 20F55, 52B40

Abstract. If Δ is a polytope in real affine space, each edge of Δ determines a reflection in the perpendicular bisector of the edge. The exchange group W(Δ) is the group generated by these reflections, and Δ is a (Coxeter) matroid polytope if this group is finite. This simple concept of matroid polytope turns out to be an equivalent way to define Coxeter matroids. The Gelfand-Serganova Theorem and the structure of the exchange group both give us information about the matroid polytope. We then specialize this information to the case of ordinary matroids; the matroid polytope by our definition in this case turns out to be a facet of the classical matroid polytope familiar to matroid theorists.

Keywords: matroid polytope, Coxeter matroid, exchange group


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