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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 Γcell. We apply the previous
terminology concerning P to Γ. Since Γcells 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 MichiganDearborn.
 Fall '04
 IgorDolgachev
 Algebra, Geometry, The Land

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