Assume that b is nondegenerate indenite and w is of

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Unformatted text preview: itting an ordered set of generators S of order 2 with defining relations (ss )m(s,s ) = 1, s, s ∈ S, where m(s, s ) is the order of the product ss (the symbol ∞ if the order is infinite). The pair (W, S ) is called a Coxeter system. The Coxeter graph of (W, S ) is the graph whose vertices correspond to S and any two different vertices are connected by m(s, s ) − 2 edges or by an edge labeled with m(s, s ) − 2 or a thick edge if m(s, s ) = ∞. We say that (W, S ) is irreducible if the Coxeter graph is connected. One proves the following theorem (see [112]). Theorem 2.5. Let P be a nondegenerate Coxeter polytope of finite volume in X n and Γ(P ) be the corresponding reflection group. The pair (Γ(P ), S ), where S is the set of reflections with respect to the set of faces of P is a Coxeter system. Its Coxeter graph is equal to the Coxeter diagram of P , and the Gram matrix of P is equal to the matrix π )(s,s )∈S ×S . (2.2) (− cos m(s, s ) The converse is partially true. The following facts can be found in [14]. Let (W, S ) be an irreducible Coxeter system with no m(s, s ) equal to ∞. One considers the linear space V = RS and equips it with a symmetric bilinear form B defined by π . (2.3) B (es , es ) = − cos m(s, s ) Assume that B is positive definite ((W, S ) is elliptic ). Then W is finite and isomorphic to a reflection group Γ in the spherical space S n , where n + 1 = #S − 1. The corresponding Γ-cell can be taken as the intersection of the sphere with the simplex in Rn+1 with facets orthogonal to the vectors es . Assume that B is degenerate and semipositive definite ((W, S ) is parabolic ). Then its radical V0 is one-dimensional and is spanned by a unique vector v0 = as = 1. The group W acts naturally as es satisfying as > 0 for all s and as a reflection group Γ in the affine subspace E n = {φ ∈ V ∗ : φ(v0 ) = 1} with the associated linear space (V /V0 )∗ . The corresponding Γ-cell has n + 1 facets orthogonal to the vectors es and is a simplex in affine space. Assume that B is nondegenerate, indefinite and W is of cofinite volume in the orthogonal group of B ((W, S ) is hyperbolic ). In this case the signature of B is equal to (n − 1, 1) and C = s∈S R+ es is contained in one of the two connected components of the set {x ∈ E : B (x, x) < 0}. Let H n be the hyperbolic space equal REFLECTION GROUPS IN ALGEBRAIC GEOMETRY 13 to the image of this component in P(E ). Then the action of W in H n is isomorphic ¯ to a reflection group Γ. A Γ-cell can be chosen to be the image of the closure C of C in P(E ). In all cases the Coxeter diagram of a Γ-cell coincides with the Coxeter graph of (W, S ). We will call the matrix given by (2.3) the Gram matrix of (W, S ). Remark 2.6. Irreducible reflection groups in S n and E n correspond to elliptic or parabolic irreducible Coxeter systems. Hyperbolic Coxeter systems define Coxeter simplices in H n of finite volume. Their Coxeter diagrams are called quasi-Lanner [120] or hyperbolic [14]. They are characterized by the condition that each of its proper subdiagrams is either elliptic or...
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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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