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

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

1 ~p 94 1 (abc) 1 1 2 -1

1 ~p 94 1 (abc) 1 1 2 -1 where 1 and (abe) denote the odd order conjugacy classes ((p is the modular character for the standard 2-dimensional representation of S 3 which is a quotient of S 4 and hence W(D4)). Thus, summing the third row in table 4.3 we obtain Richardson : 5G(Xp -- 16+12~ --- 0 mod 2.x(G) On the other hand, the maximal isotropy subgroups have been determined by Field and Table 4.4 Maximal Isotropy Subspaces type orbit size isotropy group modular perm. char. (0, 0, 0, u) 4 S 3 x (Z/2Z) 2 2 + (p (0, 0, u, u) 12 (Z/2Z) 3 4 + 4(p (u, u, u, u) 4 S 4 2 + (p (u, u, u, -u) 4 S 3 x (7/27)2 2 + ¢p (here orbit size refers to the orbits of branches not half-branches). Thus, adding the last column in table 4.4, we see that the sum ~ Xi of the characters for the nontrivial orbit of branches with maximal isotropy subgroups satisfies ~G(~Xi~) - 10+7q~ - (p ~ 0 mod 2.~(G) Thus, by corollary 6 there exist other nontrivial branches, which must be submaximal. Furthermore, on the subspace {(u, u, u, t): u, t ~ $1}, which is the fixed point space of S 3, we obtain the restiction of F F'(u, t, X) = (u 3 + 0:u2t + Xu, t 3 + Xt + ctu 3) We already know that it contains the trivial branch and the branches of types (u, u, u, u) and (u, U, U, -U). To see that there is another, we compute j~'). For F', wt(u, t, X) -- (I, 1, 2) so s' = 4, Sp' --- 2, and j0:') ~- dim rood 2 = 1; thus the number of nontrivial branches is congruent to j(F') -- 1 mod 2. Hence, there must be a third nontrivial branch. This branch has an orbit of 16 branches with modular character = 6 + 5(p. In this case the number of nontrivial branches --- 40 -- dim )(F)6 ; hence by corollary 2, we obtain a complete description of the permutation representation Vp~ :, C ~) ()(F)6 ® q~).

Example 4.5: 95 Let G = Z/mE act on ~ ~, I:l 2 by ~(z) = ~k.z, where ~ is a primitive m-th root of unity. To describe the invariant functions and equivariant vector fields, we use complex valued smooth functions; then the ring of invariant smooth functions is generated by z.2, z m, ~m, and ~.. Also, the module of equivaxiant vector fields is generated by - ~ ~m-1 ~ and zm-lff---~ z~, ~z' " The generic bifurcation germ is given in (z, ~)-coordinates by F(z, ~) = (im- 1 + Xz, z m-1 + X~) We assign weights wt(z, ~) = (1,1) and wt(X) = m - 2. Then Sp = 2(m - 1) - 2 - wt(2L) = m - 2. Then, (4.6) 5(F)Gm_ 2 = Q0~)Gm_ 2 m odd m even If m is odd then by corollary 4 there are either 0 or 2 orbits. However, there is already one trivial orbit; hence, there must be exactly 2 orbits one of which is nontrivial. For the case of m even, it is necessary to do more work and compute the signature to determine the number of orbits of branches. The Jacobian algebra is computed using ~m-1 + Xz, zm-1 + X~, and the 2 × 2 minors of Q (m- 1)~ m-2 X z X (m - 1)z m-2 1 which together give (4.7) zm-1, ~m-l, zX, 2~,, (m - 1)2(z2) m-2 - 2 The top weight space is generated by ~2. A basis for 5(F)m_ 2 = Q(F)m_ 2 is given by: (4.8) {X, Ur, s = zr~ s + zS~ r, Vr, s = i(zr~ s - zS~r); r + s -- m - 2 }. Using (4.7), we see that since z m-l, ~m-1 • J(F), then (4.8) is an orthogonal basis with respect to the multiplication pairing. Also, the multiplication pairing gives a positive multiple of ~2 for cach element of this basis so the pairing is positive definite. Thus, by corollary 2, the

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