Then the vectors vi are either linearly independent

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Unformatted text preview: ections in Γ contain x0 . Therefore each Γ-cell is a polyhedral cone with vertex at x0 . In the case when X n = E n , we can use x0 to identify E n with its linear space V and the group Γ with a reflection group in S n−1 . The same is true if X n = S n . If X n = H n and x0 ∈ H n (resp. x0 lies on the 10 IGOR V. DOLGACHEV absolute), then by considering the orthogonal subspace in Rn+1 to the line defined by x0 we find an isomorphism from Γ to a reflection group in S n−1 (resp. E n−1 ). Any convex polytope of finite volume in E n or H n is nondegenerate in the sense that its faces do not have a common point and the unit norm vectors of the faces span the vector space V . A spherical convex polytope is nondegenerate if it does not contain opposite vertices. Theorem 2.2. Let Γ be an irreducible reflection group in X n with nondegenerate Γ-cell P . If X n = S n , then Γ is finite and P is equal to the intersection of S n with a simplicial cone in Rn+1 . If X n = E n , then Γ is infinite and P is a simplex in E n. This follows from the following simple lemma ([14], Chapter V, §3, Lemma 5): Lemma 2.3. Let V be a real vector space with positive definite symmetric bilinear form (v, w) and let (vi )i∈I be vectors in V with (vi , vj ) ≤ 0 for i = j . Assume that the set I cannot be nontrivially split into the union of two subsets I1 and I2 such that (vi , vj ) = 0 for i ∈ I1 , j ∈ I2 . Then the vectors vi are either linearly independent or span a hyperplane and a linear dependence can be chosen of the form ai vi with all ai positive. The classification of irreducible nondegenerate Coxeter polytopes, and hence irreducible reflection groups Γ in S n and Rn with nondegenerate Γ-cell, was given by Coxeter [27]. The corresponding list of Coxeter diagrams is given in Table 1. Here the number of nodes in the spherical (resp. euclidean) diagram is equal to the subscript n (resp. n + 1) in the notation. The number n is equal to the rank of the corresponding Coxeter polytope. We will refer to diagrams from the first (resp. second) column as elliptic Coxeter diagrams (resp. parabolic Coxeter diagrams ) of rank n. Our classification of plane reflection groups in section 1.1 fits in this classification: ˜ ˜ ˜ ˜ ˜ (2, 4, 4) ←→ C2 , (2, 3, 6) ←→ G2 , (3, 3, 3) ←→ A2 , (2, 2, 2, 2) ←→ A1 × A1 . The list of finite reflection groups not of type H3 , H4 , I2 (m), m = 6 (I2 (6) is often denoted by G2 ) coincides with the list of Weyl groups of simple Lie algebras of the corresponding type An , Bn or Cn , G2 , F4 , E6 , E7 , E8 . The corresponding Coxeter diagrams coincide with the Dynkin diagrams only in the cases A, D, and E . The second column corresponds to affine Weyl groups. The group of type An is the symmetric group Σn+1 . It acts in the space (2.1) V = {(a1 , . . . , an+1 ) ∈ Rn+1 : a1 + . . . an+1 = 0} with the standard inner product as the group generated by reflections in vectors ei − ei+1 , i = 1, . . . n. The group of type Bn is isomorphic...
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