Hence for any g om there exists s refk m such that

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Unformatted text preview: is also easy to compute the determinant of the Gram matrix to obtain that the absolute value of the discriminant of a nondegenerate lattice Ep,q,r is equal to |pqr − pq − pr − qr |. In particular, it is a unimodular lattice if and only if (p, q, r ) = (2, 3, 5) or (2, 3, 7). One can also compute the discriminant group of a nondegenerate lattice Ep,q,r (see [17]). Every subgroup of M is considered as a lattice with respect to the restriction of the quadratic form (sublattice ). The orthogonal complement of a subset S of a lattice is defined to be the set of vectors x in M such that (x, s) = 0 for all s ∈ S . It is a primitive sublattice of M (i.e. a subgroup of M such that the quotient group is torsion-free). Also one naturally defines the orthogonal direct sum M ⊥ N of two (and finitely many) lattices. The lattice M of rank 2 defined by the matrix ( 0 1 ) is denoted by U and is called 10 the hyperbolic plane. For any lattice M and an integer k we denote by M (k) the lattice obtained from M by multiplying its quadratic form by k. For any integer k we denote by k the lattice of rank 1 generated by a vector v with (v, v ) = k. REFLECTION GROUPS IN ALGEBRAIC GEOMETRY 21 The following theorem describes the structure of unimodular indefinite lattices (see [99]). Theorem 4.4. Let M be a unimodular lattice of indefinite signature (p, q ). If M is odd, then it is isometric to the lattice Ip,q = 1 p ⊥ −1 q . If M is even, then p − q ≡ 0 mod 8 and M or M (−1) is isometric to the lattice IIp,q , p < q, equal to q −p the orthogonal sum U p ⊥ E8 8 . 4.3. Reflection group of a lattice. Recall that the orthogonal group of a nondegenerate symmetric bilinear form on a finite-dimensional vector space over a field of characteristic = 2 is always generated by reflections. This does not apply to orthogonal groups of lattices. Let M be a nondegenerate quadratic lattice. A root vector in M is a primitive vector α with (α, α) = 0 satisfying (4.1). A root vector α defines a reflection rα : x → x − 2(α, x) α (α, α) in V = MR which leaves M invariant. Obviously any vector α with (α, α) = ±1 or ±2 is a root vector. Suppose (α, α) = 2k. The linear function M → Z, x → 2(α,x)) (α,α defines a nontrivial element from M ∗ /M of order k. Thus k must divide the order of the discriminant group. In particular, all root vectors of a unimodular even lattice satisfy (α, α) = ±2. We will be interested in positive definite lattices or hyperbolic lattices M of signature (n, 1). For such a lattice we define the reflection group Ref(M ) of M as the subgroup of O(M ) generated by reflections rα , where α is a root vector with (α, α) > 0. We denote by Refk (M ) its subgroup generated by reflections in root vectors with k = (α, α) (the k-reflection subgroup ). We set Ref−k (M (−1)) = Refk (M ). Each group Refk (M ) is a reflection group in corresponding hyperbolic or spherical space. Suppose M is of signature (n, 1). Let O(...
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