1 one can choose a vector v with b v v 0 the

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Unformatted text preview: parabolic. If no parabolic subdiagram is present, then the simplex is compact and the diagram is Lanner or compact hyperbolic. The complete list of hyperbolic Coxeter diagrams can be found in [120] or [58]. Example 2.7. Let W (p, q, r ), 1 ≤ p ≤ q ≤ r, be the Coxeter group with Coxeter graph of type Tp,q,r given in Figure 7. • • q ... • ⎧• ⎪ ⎪ ⎪ ⎪ ⎪• ⎪ ⎪ ⎨ p. . ⎪. ⎪ ⎪ ⎪ ⎪• ⎪ ⎪ ⎩ • r ... • • • Figure 7. Then • W (p, q, r ) is of elliptic type if and only if (p, q, r ) = (1, q, r ), (2, 2, r ), (2, 3, 3), (2, 3, 4), (2, 3, 5); • W (p, q, r ) is of parabolic type if and only if (p, q, r ) = (2, 4, 4), (2, 3, 6), (3, 3, 3); • W (p, q, r ) is of hyperbolic type if and only if (p, q, r ) = (3, 4, 4), (2, 4, 5), (2, 3, 7). 3. Linear reflection groups 3.1. Pseudo-reflections. Let E be a vector space over any field K . A pseudoreflection in E is a linear invertible transformation s : E → E of finite order greater than 1 which fixes pointwise a hyperplane. A reflection is a diagonalizable pseudoreflection. A pseudo-reflection is a reflection if and only if its order is coprime to the characteristic of K . Let v be an eigenvector of a reflection s of order d. Its eigenvalue η different from 1 is a dth root of unity η = 1. We can write s in the form (3.1) s ( x) = x − ( x) v 14 IGOR V. DOLGACHEV for some linear function : E → K . Its zeroes define the hyperplane of fixed points of s, the reflection hyperplane. Taking x = v we obtain (v ) = 1 − η . This determines uniquely when v is fixed; we denote it by v . A pseudo-reflection (reflection) subgroup of GL(E ) is a subgroup generated by pseudo-reflections (reflections). Assume that we are given an automorphism σ of K whose square is the identity. ¯ We denote its value on an element λ ∈ K by λ. Let B : E × E → K be a σ -hermitian form on E ; i.e. B is K -linear in the first variable and satisfies B (x, y ) = B (y, x). Let U(E, B, σ ) be the unitary group of B , i.e. the subgroup of K -linear transformations of E which preserve B . A pseudo-reflection subgroup in GL(E ) is a unitary pseudo-reflection group if it is contained in a unitary group U(E, B, σ ) for some σ, B and for any reflection (3.1) one can choose a vector v with B (v, v ) = 0. The additional condition implies that (3.2) v ( x) = (1 − η )B (x, v ) for all x ∈ V. B (v, v ) In particular, the vector v is orthogonal to the hyperplane −1 (0). v Finite reflection groups are characterized by the following property of its algebra of invariant polynomials ([14], Chapter V, §5, Theorem 4). Theorem 3.1. A finite subgroup G of GL(E ) of order prime to char(K ) is a reflection group if and only if the algebra S (E )G of invariants in the symmetric algebra of E is isomorphic to a polynomial algebra. In the case K = C this theorem was proven by Shephard and Todd [102], and in the case of arbitrary characteristic but for groups generated by reflectio...
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