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5 years ago

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

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

Now, we can compute the

Now, we can compute the modular characters via the following theorem. Theorem 1: Suppose that F : IRn+l,o ---. [I n ,0 is G-equivariant homogeneous with initial part F O. Then, 88 1) Zp = 1 + sigG( 5(F0)sp ) 2) 7

in particular, Zp~ = 1 + ~ iii) Vp¢ - ¢ ~ C~fFO)sp~ ) as complex G-representations 89 Remark: It appears to be more than coincidence that for the generic bifurcation problems which this author has examined, the multiplication pairing on J(F0)sp is positive definite (for at least some open subset of the values of the moduli) so that ii) and iii) of corollary 2 apply. It would be interesting to know whether this always holds, and if so, to understand what is the underlying reason. following. As further corollaries of the theorem and these corollaries we may conclude the Corollary 3: If F is as in theorem I then: i) ff wtO0 is odd then ~b = O, so that the modular characters for the permutation representations on the set of haft branches where ~ < 0 respectively. ~ > 0 are the same; ii) if s is odd then ~d = O, so that the modular characters for the permutation representations on the set of half branches where sign(det(gradx(F))) < O, respectively > O, are the same. Remark: At the very least, corollary 3 allows us to conclude that there are the same number of branches of each sign. Corollary 4: /f F is as in theorem I and G has odd order then: the number of G-orbits in B = sig(XF0)spG) + 1 < dirno~(XF0)sp G) + 1 (here .~]~0)Sp G denotes the subspace of elements invafiant under the G-action). Remark: Even if G itself does not have odd order, we may still use corollaries 2 and 4 to obtain information about the orbit structure by restricting the action to a subgroup of G of odd order. Lastly, we explicitly restate the conclusions of these results as they apply to equivariant morsifications. Let H(x,X): [in+l,0 ~ ~,0 be an equivariant morsification of the germ h which has an isolated singularity at 0. We shall say that H is semi-weighted homogeneous if gradx(H) is; and that it is semi-weighted homogeneous as a morsification if gradx(H) is semi- weighted homogeneous for bifurcation equivalence. We let F = gradx(H). H defines a family of germs H~ : U ~ $1 for some neighborhood U of 0 and for I k I < e. We let C = C+ u C_, where C+ = crit(H~.), C_ = crit(H~,), and "or it" denotes the set

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