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Unformatted text preview: extending the
bilinear form by linearity.
Recall some terminology in the theory of integral quadratic forms stated in terms
of lattices. The signature of a lattice M is the Sylvester signature (t+ , t− , t0 ) of the
corresponding real quadratic form on V = MR . We omit t0 if it is equal to zero.
A lattice with t− = t0 = 0 (resp. t+ = t0 = 0) is called positive deﬁnite (resp.
negative deﬁnite ). A lattice M of signature with (1, a) or (a, 1) where a = 0 is
called hyperbolic (or Lorentzian ).
All lattices are divided into two types: even if the values of its quadratic form
are even and odd otherwise.
Assume that the lattice M is nondegenerate; that is t0 = 0. This ensures that
the map
(4.4) ιM : M → M ∗ = HomZ (M, Z), m → (m, ?) 20 IGOR V. DOLGACHEV is injective. Since M ∗ is an abelian group of the same rank as M , the quotient
group
DM = M ∗ /ι(M )
is a ﬁnite group (the discriminant group of the lattice M ). Its order dM is equal to
the absolute value of the discriminant of M deﬁned as the determinant of a Gram
matrix of the symmetric bilinear form of M . A lattice is called unimodular if the
map (4.4) is bijective (equivalently, if its discriminant is equal to ±1).
Example 4.2. Let M be the lattice deﬁning an integral structure on a ﬁnite reﬂection group from Example 4.1. It is an even positive deﬁnite lattice for the groups of
types A, D, E and odd positive deﬁnite lattice for groups of type Bn , F4 , G2 . These
lattices are called ﬁnite root lattices of the corresponding type.
Example 4.3. Let Γ be an irreducible linear reﬂection group in V admitting an
integral structure M . It follows from (4.2) that, after rescaling the inner product
in V , we may assume that M is a lattice in V with MR = V . For example, consider
the group Γ = W (p, q, r ) from Example 2.7 as a linear reﬂection group in Rn , where
n = p + q + r − 2. The unit vectors ei of a fundamental Coxeter polytope satisfy
π
(ei , ej ) = −2 cos mij , where mij ∈ {1, 2, 3}. Thus, rescaling the quadratic form in
V by multiplying its values by 2, we ﬁnd fundamental root vectors αi such that
(αi , αj ) ∈ Z. The lattice M generated by these vectors deﬁnes an integral structure
of Γ. The Gram matrix G of the set of fundamental root vectors has 2 at the
diagonal, and 2In − G is the incidence matrix of the Coxeter graph of type Tp,q,r
from Example 2.7. We denote the lattice M by Ep,q,r . One computes directly the
signature of M to obtain that Ep,q,r is nondegenerate and positive deﬁnite if and
only if Γ is a ﬁnite reﬂection group of type A, D, E (r = 1(An ) or r = p = 2(Dn )
or r = 2, p = 3, q = 3, 4, 5(E6 , E7 , E8 )).
The lattice Ep,q,r is degenerate if and only if it corresponds to a parabolic re⊥
⊥
˜˜˜
ﬂection group of type E6 , E7 , E8 . The lattice Ep,q,r is of rank 1 and Ep,q,r /Ep,q,r
is isomorphic to the lattice Ep−1,q,r , Ep,q−1,r , Ep,q,r−1 , respectively.
In the remaining cases Ep,q,r is a hyperbolic lattice of signature (n − 1, 1).
It...
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 Fall '04
 IgorDolgachev
 Algebra, Geometry, The Land

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