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
sasha@lanczos.ma.umist.ac.uk

Department of Mathematics, Rutgers University, New Brunswick, NJ 08903, USA
igelfand@math.rutgers.edu

Department of Mathematics, University of Florida, Gainesville, FL 32611, USA
white@math.ufl.edu

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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