This proves i iii dene a coxeter polytope to be a

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Unformatted text preview: ) of Γ. Its faces are called walls. Two cells which share a common wall are called adjacent. The corresponding reflection switches the adjacent cells. This easily shows that the group Γ permutes transitively the cells. Also it shows that any hyperplane from H is the image of a wall of P under an element of the group Γ(P ) generated by the reflections with respect to walls of P . Thus Γ = Γ(P ). It is clear that the orbit of each point intersects a fixed cell P . The proof that no two interior points of P belong to the same orbit follows from the last assertion of the theorem. Its proof is rather complicated, and we omit it (see [120], Chapter V, Theorem 1.2). Let H, H be two hyperplanes bounding P for which the angle ∠(H − , H − ) is defined and is not zero. The corresponding unit vectors e, e span a plane in the vector space V associated to X n , and the restriction of the symmetric bilinear form to the plane is positive definite. The subgroup of Γ generated by the reflections rH , rH defines a reflection subgroup in Π. Thus the angle must be of the form rπ for some rational number r . If r is not of the form 1/m for some integer m, then Γ contains a reflection with respect to a hyperplane intersecting the interior of P . By definition of P this is impossible. This proves (i)-(iii). Define a Coxeter polytope to be a convex polytope P in which any two faces are either divergent or form the angle equal to zero or π/m for some positive integer m − (or ∞). Let (ei )∈I be the set of unit vectors corresponding to the halfspaces Hi defining P . The matrix G(P ) = ((ei , ej ))(i,j )∈I ×I is the Gram matrix of P , and its rank is the rank of the Coxeter polytope. The polytope P is called irreducible if its Gram matrix is not equal to the nontrivial direct sum of matrices. One can describe the matrix G(P ) via a certain labeled graph, the Coxeter diagram of P . Its vertices correspond to the walls of P . Two vertices corresponding to the hyperplanes with angle of the form π/m, m ≥ 3, are joined with an edge labeled with the number m − 2 (dropped if m = 3) or joined with m − 2 nonlabeled edges. Two vertices corresponding to parallel hyperplanes (i.e. forming the zero angle) are joined by a thick edge or an edge labeled with ∞. Two vertices corresponding to divergent hyperplanes are joined by a dotted edge. Obviously an irreducible polytope is characterized by the condition that its Coxeter diagram is a connected graph. Let Γ be a reflection group in X n and P be a Γ-cell. We apply the previous terminology concerning P to Γ. Since Γ-cells are transitively permuted by Γ, the isomorphism type of the Gram matrix does not depend on a choice of P . In particular, we can speak about irreducible reflection groups. They correspond to Gram matrices which cannot be written as a nontrivial direct sum of their submatrices. Suppose all elements of Γ fix a point x0 in X n (or on the absolute of H n ). Then all mirror hyperplanes of re...
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This note was uploaded on 02/24/2012 for the course MATH 285 taught by Professor Igordolgachev during the Fall '04 term at University of Michigan-Dearborn.

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