Obviously every quadratic lattice is obtained in this

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Unformatted text preview: mitive vector in M (i.e. M/ZαH is torsion-free). We call such a vector a root vector associated to H . The root vectors corresponding to the faces of a fundamental polytope are called the fundamental root vectors. A root vector is uniquely defined by H up to multiplication by −1. Using the primitivity property of root vectors it is easy to see that, for all x ∈ M , (4.1) 2(x, α) ∈ (α, α)Z. In particular, if α, β are not perpendicular root vectors, then 2(α, β )/(β, β ) and 2(α, β )/(α, α) are nonzero integers, so that the ratio (β, β )/(α, α) is a rational number. If the Coxeter diagram is connected, we can fix one of the roots α and multiply the quadratic form (x, x) on V by (α, α)−1 to assume that (β, β ) ∈ Q for all root vectors. This implies that the Gram matrix (ei , ej ) of a basis of M has REFLECTION GROUPS IN ALGEBRAIC GEOMETRY 19 entries in Q. Multiplying the quadratic form by an integer, we may assume that it is an integral matrix; hence (4.2) (x, y ) ∈ Z, for all x, y ∈ M. Example 4.1. Let Γ be an irreducible finite real reflection group whose Dynkin diagram does not contain multiple edges (i.e. of types A, D, E ). Let (α1 , . . . , αn ) be the unit norm vectors of its fundamental Coxeter polytope and let M ⊂ V be the span of these vectors. If we multiply the inner product in V by 2, we obtain π (αi , αj ) = −2 cos mij ∈ {0, 2, −1}. Hence the reflections rαi : x → x − (x, αi )αi leave M invariant. Thus Γ admits an integral structure and the Gram matrix of its basis (α1 , . . . , αn ) is equal to twice the matrix (2.2). Note that (αi , αi ) = 2, i = 1, . . . , n. Let Γ be of type Bn and es , s = 1, . . . , n, be the unit normal vectors defined by a fundamental polytope. We assume that m(n, n − 1) = m(n − √ n) = 4 and other 1, m(s, s ) take values in {1, 2, 3}. Let αi = ei if i = n and αn = 2en . It is easy to i) see now that 2(x,αi ) ∈ Z for any x in the span M of the αi ’s. This shows that M (αi ,α defines an integral structure on Γ and (x, y ) ∈ Z for any x, y ∈ M . We have (4.3) (αi , αi ) = 1 if i = n, 2 otherwise. We leave it to the reader to check that the groups of type F4 and G2 = I2 (6) also admit an integral structure. However, the remaining groups do not. 4.2. Quadratic lattices. A (quadratic) lattice is a free abelian group M equipped with a symmetric bilinear form with values in Z. The orthogonal group O(M ) of a lattice is defined in the natural way as the subgroup of automorphisms of the abelian group preserving the symmetric bilinear form. More generally one defines in an obvious way an isometry or isomorphism of lattices. Let V be a real vector space equipped with a symmetric bilinear form (x, y ) and (ei )i∈I be a basis in V such that (ei , ej ) ∈ Z for all i, j ∈ I . Then the Z-span M of the basis is equipped naturally with the structure of a quadratic lattice. We have already seen this construction in the beginning of the section. The orthogonal group O(M ) coincides with the group introduced there. Obviously every quadratic lattice is obtained in this way by taking V = MR = M ⊗Z R and...
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