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
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
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
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) 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))
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
Suppose M is of signature (n, 1). Let
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