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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 three-dimensional reﬂection representation. This and other examples of appearance of ﬁnite unitary complex reﬂection groups as the G-equivariant 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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