5 34 k6 108 9 5 35 e6 72 6 6 36 e7 8 9 7 37 e8 192 9

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Unformatted text preview: ns of order 2, it was proven by C. Chevalley [19]. We will be concerned with the case where K = R and σ = idE or K = C and σ is the complex conjugation. In the real case a reflection is necessarily of order 2. Let G be a finite reflection subgroup of GL(E ). By taking some positive definite symmetric bilinear (resp. hermitian) form and averaging it, we see that G is conjugate in GL(E ) to a unitary reflection group. In fact, under isomorphism from E to the standard euclidean (resp. unitary) space Rn (resp. Cn ), the group G is isomorphic to a reflection subgroup of O(n) (resp. U(n)). In the real case G becomes isomorphic to a reflection group in S n−1 and hence is isomorphic to the product of irreducible reflection spherical groups. An example of an infinite real reflection group in Rn+1 is a reflection group in the hyperbolic space H n . It is an orthogonal reflection group with respect to a symmetric bilinear form of signature (n, 1). 3.2. Finite complex linear reflection groups. They were classified by Shephard and Todd [102]. Table 2 gives the list of irreducible finite linear reflection groups (in the order given by Shephard-Todd; see also the table in [62], p. 166). The last column in the table gives the degrees of the generators of the algebra S (E )G . The group G(m, p, n) is equal to the semi-direct product A(m, n, p) Σn , where A(m, n, p) is a diagonal group of n × n-matrices with mth roots of unity at the diagonal whose product is an (m/p)th root of unity. The semi-direct product is defined with respect to the action of Σn by permuting the columns of the matrices. Groups 3-22 are some extensions of binary polyhedral groups (i.e. finite subgroups of SL(2, C)). REFLECTION GROUPS IN ALGEBRAIC GEOMETRY 15 Table 2. Finite complex reflection groups Number Name Order dim E 1 An = Σn+1 (n + 1)! n 2 G(m, p, n) mn n!/p n 3 m m 1 4 3[3]3 24 2 5 3[4]3 72 2 6 3[6]2 48 2 7 < 3, 3, 3 >2 144 2 8 4[3]4 96 2 9 4[6]2 192 2 10 4[4]3 288 2 11 < 4, 3, 2 >12 576 2 12 GL(2, 3) 48 2 13 < 4, 3, 2 >2 96 2 14 3[8]2 144 2 15 < 4, 3, 2 >6 288 2 16 5[3]5 600 2 17 5]6]2 1200 2 18 5[4]3 1800 2 19 < 5, 3, 2 >30 3600 2 20 3[5]3 360 2 21 3[10]2 720 2 22 < 5, 3, 2 >2 240 2 23 H3 120 3 24 J3 (4) 336 3 25 L3 648 3 26 M3 1296 3 27 J3 (5) 2160 3 28 F4 1152 4 29 N4 7680 4 30 H4 14,440 4 31 EN4 64 · 6! 4 32 L4 216 · 6! 4 33 K5 72 · 6! 5 34 K6 108 · 9! 5 35 E6 72 · 6! 6 36 E7 8 · 9! 7 37 E8 192 · 9! 8 Degrees 2, 3, . . . , n + 1 m, m + 1, . . . , (n − 1)m, mn/p m 4,6 6,12 4,12 12,12 8,12 8,24 12,24 24,24 6,8 8,12 6,24 12,24 20,30 20,60 60,60 60,60 12,30 12,60 12,20 2,6,10 4,6,14 6,9,12 6,12,18 6,12,30 2,6,8,12 4,8,12,20 2,12,20,30 8,12,20,24 12,18,24,30 4,6,10,12,18 4,6,10,12,18 2,5,6,8,9,12 2,6,8,10,12,14,18 2,8,12,14,18,20,24,30 All real spherical irreducible groups are in the list. We have seen already the groups of types An , E6 , E7 , E8 , H3 , H4 , F4 . The groups of type Bn are the groups G(2, 1, n). The groups of type D...
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