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.
KeywordsSimplicial Complex Parabolic Subgroup Coxeter Group Isotropy Subgroup Reflection Group
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