5 years ago

SN~ (~6) lff 2It 3k_~ , 5 ",,x_J {b)

SN~ (~6) lff 2It 3k_~ , 5 ",,x_J {b)


permutation representation is given by ¢ ~ (XF)m_ 2 ® ¢) and by (4.6) there are exactly three orbits, two of which are nontrivial. Example (4.9) 96 Let W(D 3) again denote the Weyl group which is generated by the group S 3 of permutations on {x 1, x 2, x 3} together with the transformations x i I-~ -1- x i of determinant = 1. This group has a natural action on ~3. We let G denote the subgroup generated by the subgroup Z 3 c S 3 and the transformations x i b-¢ :t: x i of determinant = 1. Then, the generic G-equivariant bifurcation germ is given by F(x,X) = (x13 + ctx22x 1 + ~x32x 1 + ~.x I ..... x33 + OtXl2X 3 + ~x22x 3 + Xx 3) The modular characters for G are given by the 1-dimensional representations for Z 3. These have characters gi such that Zi(1) = 1 and Xi((abc)) = ~3 i, with ~3 a primitive cube root of unity. Then, the representation ofS 3 on ¢3 has modular character = ~ ~=f 1 + Xl + X2 (74) = 1). The maximal isotropy subspaces together with the size of the orbits (of branches) and the modular permutation characters are given in table 4.11. Table 4.11 Maximal !s0tropy Subspaces type orbit size isotropy group modular penn. char. (0, 0, a) 3 Z/2E ¥ (0, a, a) 6 l/2Z 2xF (a, a, a) 4 13 1 + We also see that the middle weight Sp = 3.3 - 3 - 2 = 4, and the coordinate functions of F allow us to replace terms involving xi3 in Q(F) 4 ( = 5(F)4) by terms involving lower powers of x i. As in the first example, a basis is given by monomials which under the Z 3- action generate a representation. Table 4.12 X 2 Xxix j xi2xj2 xi2xjx k dim 1 6 3 3 char 1 2~g V ~g

97 Thus, we see that X q = 1 + 4V. By corollary 6, we conclude that for any values (0~, 13) for which F defines an isolated singularity, at least the (0, 0, a) and (a, a, a) orbits of branches must exist. Of course the equivariant branching lemma implies that the third orbit of branches must exist as well. Then, from theorem 1 we deduce that there are no other nontrivial branches. §5 The Non-semi-weighted Homogeneous Case In this section we extend theorem 1 to the non-weighted homogeneous case. The nonequivariant case was In'st analyzed via the results of Aoki, Fukuda, and Nishimura [AFN] and [AFN2.]. We shall give equivariant versions of their results. Let F: ~ln+l,0 ~ ~ln,0 be a smooth germ which has finite codimension for imperfect bifurcation equivalence and such that F(x,0) -- fix) is a finite map germ. Let Jac(F) denote det(dxF ). Define map germs F 1 , F 2 : ~1 n+l,0 ~ ~n+l,0 by F 1 = (F, ~..Jac(F)) and F 2 = (F, Jac(F)). By our assumption on F, they are finite map germs. Also, F i has local algebra Q(Fi) -- Cx,~/I(Fi). Note that Q(F 2) is what we referred to as the bifurcation algebra in §1. Then, Aoki, Fukuda, and Nishimura relate the degrees of the mappings F 1 and F 2 with the number of branches of F-I(0) and the virtual number of branches with ~ > 0 versus < 0. By the result of Eisenbud-Levine [EL], these degrees can, in turn, be computed as the signatures of the above local algebras. These results give very nice and complete theoretical results,but they can be quite difficult to compute in many cases. For example, if the complexified germ has m branches, then these more general results require the computation of a signature on a vector space whose dimension is of order m 2. By constrast, in the semi-weighted homogeneous case, theorem 1 (excluding the group action) requires the the calculation of the signature of a quadratic form on an m-dimensional vector space. Although the equivariant versions of these results which we shall give are stated in terms of G-signatures, the use of the reduced methods of §3 with these results allow us to obtain representation theoretic information about the branches without computing a signature. We consider a representation of G on ~1 n and extend it to ~i n+l by letting G act trivially on the last factor. Let F: ~n+l,0 ~ ~n,0 be a G-equivariant smooth germ which has an isolated singularity at 0 and is the bifurcation germ of a finite map germ F(x,0) = f(x). Define F 1, and F 2 as above. Because the G-action on the last factor of ~! n+l is the trivial representation, Jac(F) (-- det(dxF)) is G-invariant. As ~, is G-invariant, the germs F i are G- equivariant and the germs Jac(F i) = det(d(x,~,)F i) are G-invariant. Thus, we may choose G- invariant linear functionals ~i : Q(Fi) ~ ~1 with ~i(Jac(Fi)) > 0. Then, by local duality, the multiplication pairings on Q(Fi) defined by 9i(a,b) = ~i(a'b) are nonsingular and, by

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