Coxeter Matroids

  • Alexandre V. Borovik
  • I. M. Gelfand
  • Neil White
Part of the Progress in Mathematics book series (PM, volume 216)


In this chapter we develop the general theory of Coxeter matroids for an arbitrary finite Coxeter group, thus generalizing most of the results from Chapters 1 and 3. The keystone to the whole theory is the Gelfand—Serganova Theorem which interprets Coxeter matroids as Coxeter matroid polytopes (Theorem 6.3.1). As we shall soon show (Theorem 6.4.1), the latter can be defined in a very elementary way:

Let Δ be a convex polytope. For every edge [ α, β] of Δ, take the hyperplane that cuts the segment [α, β] at its midpoint and is perpendicular to [α, β]. Let W be the group generated by the reflections in all such hyperplanes. Then W is a finite group, if and only if Δ is a Coxeter matroid polytope.


Simplicial Complex Parabolic Subgroup Coxeter Group Isotropy Subgroup Reflection Group 
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Copyright information

© Birkhäuser Boston 2003

Authors and Affiliations

  • Alexandre V. Borovik
    • 1
  • I. M. Gelfand
    • 2
  • Neil White
    • 3
  1. 1.Department of MathematicsUMISTManchesterUK
  2. 2.Department of MathematicsRutgers UniversityPiscatawayUSA
  3. 3.Department of MathematicsUniversity of FloridaGainesvilleUSA

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