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Unformatted text preview: , G2 can be obtained from those of types A, D, E .
O • • • . . . • • • vv
v
vA
• 2k−1
r
• • • . . . • • • rr
r
• • • ... • • • • O • • vv
v
v•
•
r
• • rr
r E6 •••• F4 Bk O • vv
v
v • ... • • • • D
•
r
k+1
• rr
r
• • • ... • • • • Bk OO t
t
• ttt
•
O • ttt D4
t
•t
• • G2 Figure 14.
Remark 9.4. The appearance of the Dynkin diagrams of type Bn , F4 , G2 in the
theory of simple singularities was ﬁrst noticed by P. Slodowy [106], and from a
diﬀerent but equivalent perspective in the work of Arnol’d on critical points on
manifolds with boundary [6], [7]. In the theory of simple surface singularities over
nonalgebraically closed ﬁeld, they appear in [73].
Example 9.5. Let (X, 0) be a simple surface singularity of type E6 given by
2
3
4
equation z1 + z2 + z3 = 0. Consider the group G generated by the symmetry g of
order 3 given by (z1 , z2 , z3 ) → (z1 , η3 z2 , z3 ), where η3 = e2πi/3 . A monomial basis
2
2
of the jacobian algebra is (1, z2 , z3 , z3 , z2 z3 , z2 z3 ). By Theorem 9.2, we have
MC = (MC )χ ⊕ (MC )χ ,
¯
where χ(g )(x) = η3 x. The characteristic polynomial of g in MC is equal to (t2 +
t + 1)3 . We have
O(M ) = W (E6 ) (Z/2Z),
where the extra automorphism is deﬁned by the symmetry of the Coxeter diagram.
Since g is of order 3, its image w in O(E6 ) belongs to W (E6 ). The classiﬁcation
of elements of order 3 in the Weyl group W (E6 ) shows the conjugacy class of w
corresponds to the primitive embedding of lattice A3 → E6 so that w acts as the
2
product c1 c2 c3 of the Coxeter elements in each copy of A2 . It is known that the
centralizer of w is a maximal subgroup of W (E6 ) of order 648.13 This group is
13 Not to be confused with another maximal subgroup of W (E ) of the same order which is
6
realized as the stabilizer subgroup of the sublattice A3 .
2 REFLECTION GROUPS IN ALGEBRAIC GEOMETRY 51 isomorphic to the unitary complex reﬂection group L3 (No. 25 in the list), and
(MC )χ is its threedimensional reﬂection representation.
This and other examples of appearance of ﬁnite unitary complex reﬂection groups
as the Gequivariant monodromy groups were ﬁrst constructed by V. Goryunov [49],
[48].
Aﬃne complex crystallographic reﬂection groups can also be realized as Gequivariant monodromy groups. We give only one example, referring for more
to [51], [50].
Example 9.6. Consider a simple elliptic singularity of type J10 with parameter
λ = 0. Its equation is given in Table 4. Consider the symmetry of order 3 deﬁned
by an automorphism g : (z1 , z2 , z3 ) → (z1 , ζ3 z2 , z3 ). A monomial basis of the
24
2
4
jacobian algebra is (1, z2 , z3 , z3 , z3 , z2 z3 , z2 z3 , z2 z3 ). We have 5 invariant monomials:
234
1, z3 , z3 , z3 , z3 . Applying Theorem 9.2 we obtain that MC = (MC )χ ⊕ (MC )χ ,
¯
both summands of dimension 5. The characteristic polynomial of g is equal to
¯
(1 + t + t2 )5 . Obviously, g leaves M ⊥ invariant, and the image w =...
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 Fall '04
 IgorDolgachev
 Algebra, Geometry, The Land

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