5 years ago

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

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

82 weight parts of these

82 weight parts of these algebras 5(F)sp, B0~)Sb , Q(F)sd. Also, we give general forms of the formulas which are valid in the non-semi-weighted homogeneous cases and provide equivariant versions of results of Eisenbud-Levine [EL] and Aoki-Fukuda-Nishimura [AFN]. In [GZ], Guzein-Zade obtained an equivariant index for gradient vector fields which gives information concerning the representation in 3),which he computes using intersection pairing information on the Milnor fiber of (the complexification of) h. Once we have obtained these results, we use them to obtain reduced sufficient conditions for the existence of branches (including submaximal branches). These sufficient conditions for the existence of branches and orbits of branches follow from purely representation-theoretic considerations on the local algebra of F, and they do not require the computation of the G-signature. This makes the computations completely practical at the expense of only obtaining partial information (see §3). For these reduced conditions, we define a homomorphism from the group of regular virtual characters X(G) G G : X(G) --, (Z/2Z) r where r is the number of odd order conjugacy classes in G. This is an equivariant analogue at the level of characters of a mod 2 degree. Via this homomorphism, we give a sufficient condition for the existence of submaximal branches (see corollary 6). Let {Xi~ : i = 1 ..... s} denote the regular characters for the orbits of branches with maximal isotropy and X~ , the regular character of the representation of G on .KF)~ (~" Q(F)~). ~p ~p If ~SG(X~ - ~Xi~) ~ 0 then there exist submaximal branches. For example, for the symmetric groups S n this condition can be verified by simply examining the parity of XStl; - ~Xi~ on odd order conjugacy classes. Secondly, for a bifurcation germ F, there are three independent mod 2 invariants j(F), b(F), and d(F) which are dimensions of certain weighted parts of the local algebra of F. We shall prove that a sufficient condition for the existence of nontrivial branches for F is (corollary 8) that (j(F), b(F), d(F) - 1) # 0 ¢ (1/27)3. Unlike the results of Field, these results only apply to Finite groups. However, we are able to use topological determinacy arguments via the semi-weighted homogeneity in place of his transversality arguments; and in the case of f'mite G, the topological degree arguments are extended using a larger collection of invariants.

83 The author's work on these questions benefitted considerably from conversations with James Montaldi, Mike Field, Mark Roberts, Ian Stewart, and Marly Golubitsky. He would also like to thank the referee for pointing out an error in the first version of this paper and other suggestions. Also, he is especially endebted to the Mathematical Institute at the University of Warwick and the organizers of the special year in Singularities and Bifurcation Theory who made this work possible. §1 §2 §3 §4 §5 §6 §7 Contents Permutation Representations, Modular Characters, and the G-signature Formulas for the Characters in the Semi-weighted Homogeneous Case Reduced Methods for the Existence of Branches Examples The Non-semi-weighted Homogeneous Case Proofs for the Permutation and Degree Characters Proofs of the Formulas for the Bifurcation Characters §1 Permutation Representations, Modular Characters, and G-signature Algebras We suppose that we are given a representation of G on [I n which is extended to act trivially on the last factor of ~n+l. We furthermore consider a G-equivariant smooth germ F(x,)0:IRn+l,0 -----} [In,0 such that: i) F-l(0) is a curve with an (algebraically) isolated singularity at 0 and ii) F(x,0) = f(x): IRn,0 -----} [in,0 is a (G-equivariant) finite map germ We let C x denote the algebra of smooth germs h: ~n,0 -----} tq which has maximal ideal m x. Likewise, Cx,~. denotes the algebra of smooth germs on [qn+l,0. For the above F we define three algebras: the Jacobian algebra ~ = Cx&/J(F) , where J(F) denotes the ideal generated by F 1, .... F n , the coordinate functions of F, together with the n x n minors of the Jacobian matrix the local algebra Q(f) = Cx/I(f) , where I(f) denotes the ideal generated by fl ..... fn, the coordinate functions of L the bifurcation algebra ~F) = Cx,~/B(F) , where B(F) denotes the ideal generated by F 1 ..... F n , the coordinate functions of F, together with det(dxF). Note: This bifurcation algebra was the one used by Aoki-Fukuda-Nishimura [AFN2] in their

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