Coxeter Matroids

Abstract

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.