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

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

90 of critical points

90 of critical points with ~.' < 0 < ~. (k and ~.' sufficiently small). Then G acts on C, C+, and C_ with modular characters X¢, Zc + ,X¢- • Also, we can write C = C O u C e , where C O , respectively C e, denotes the set of points x ~ C where H~. ( or H~,) has odd, respectively even, morse index. Again G acts on C O and C e with modular characters Zo and Xe. The modular characters for the various permutation representations are given by the following, where we let F -~ gradxfH). Corollary 5: If H(x,7~): ~ln+l,0 --+ ~,0 is an equivariant morsificafion and is semi- weighted homogeneous then i) Zc -- 2(1 + sigG(Y(F0)sp)) ii) Ze and Zo are given by where e = 1 for Ze and - 1 for Zo. 1 + sigG(Y(F0)sp) + e.sigG(Q(f0)sd) iii) /fH is semi-weighted homogeneous as a morsffication then Zc + and Xc-are given by 1 + sigG(AF0)sp) -t- sigG(q~FO)sb ) These results will be proven in §§6 and 7. Next we shall see how these results lead to simplified sufficient conditions for the existence of orbits which do not even require the computation of G-signatures. §3 Reduced Methods for the Existence of Branches We again consider F(x,~): ~n+l,0 ~ IRn,0 a G-equivariant bifurcation germ as earlier and consider several basic questions concerning the existence of branches for F-l(0). 1) Do there exist branches with submaximal isotropy ? 2) Given a maximal isotropy subgroup G' with dim Fix(G') > 1, do there exist branches with isotropy subgroup G' ? We show in this section that the results from the previous section allow us to give criteria for answering these questions. Furthermore these criteria may be stated in such a way that we do not have to compute the G-signature nor explicitly compute the Jacobian nor bifurcation algebras. In addition, the explicit use of modular characters will be reduced to a minumum. As such, the criteria give a x fast but crude wayN of checking for answers to these questions. We let X(G) denote the abelian group generated by the characters of G-representations.

91 Similarly we let X(G) denote the abelian group generated by the modular characters of G- representations. Fact 3.1 : Both of these groups are free abelian groups with ranks equal to the number of conjugacy classes of G, respectively the number of odd order conjugacy classes. Also, a set of generators for X(G) is given by the characters of the irreducible representations (this follows from classical representation theory and by a result of Brauer for modular representations, see e.g. [DP, chap. 2, 3] ). Also, there is the restriction homomorphism x(G) )~! ; RIG* This induces a surjective "reduction homomorphism" (3.2) 8 G : X(G) ~ ~G)/2.~G) ~, (E/2Z) r where r = the number of odd order conjugacy classes of G. We refer to the ~.-axis as the trivial orbit or tn'vial branch. We are interested in the orbits of nontrivial branches. If B i is an orbit of branches, then we have the regular character Zi~ of the associated permutation representation. As in §2, we let )C~ denote the regular character for J(F0)sp. The first condition that must be satisfied if {]3 i} is the complete set of nontrivial orbits can be stated in terms of ~G" CoroLlary 6 : /n the preceding situation, if {B i} is the complete set of nontriviM orbits then (3.3) ~G(Z~, - ~ ~i¢) = 0 • (7/2Z) r Remark: Analogues of (3.3) hold for Z~I; and ZQ¢ Hence, we may first find the orbits (of branches) with maximal isotropy {B' i} guaranteed by the equivariant branching lemma. Then, we ask whether there are other branches. If (13' i} have regular characters {Z'i¢} for the corresponding permutation representations, then a sufficient condition for the existence of additional orbits is that (3.4) 5G(ZJ~ - ~ Z'i¢) # 0. The reduced form of this condition often minimizes the specific information about the modular characters that is required. Also, the extra orbits which occur, together with the {I3' i} must still satisfy (3.3), which significantly aids in identifying the extra orbits. For example, for the symmetric group S n , all of the ordinary characters take only

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