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

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

Equivariant Bifurcations

Equivariant Bifurcations and Morsifications for Finite Groups by James Damon 1 ABSTRACT For a bifurcation germ F(x,~.): ~ln+l,0 ~ @n,0 which is equivariant with respect to the action of a finite group G, there are permutation actions of G on various subsets of branches of F-l(0). These sets include the set of all branches as well as the set of branches where k > 0 or < 0 or where sign(det(dxF)) > 0 or < 0. We shall give formulas for the modular characters of these permutation representations (which are the regular characters restricted to the odd order elements of G). These formulas are in terms of the representations of G on certain finite dimensional algebras associated to F. We deduce sufficient conditions for the existence of submaximal orbits by comparing the permutation representations for maximal orbits with certain representations of G. In this paper we will be concerned with describing the action of a finite group on the branches of solutions for equivariant bifurcation problems and branches of critical points for equivariant morsifications. In [G] Golubitsky asked whether the branches for generic equivariant bifurcation problems must have maximal isotropy. This was shown not to be the case by Lauterbach ILl and Chossat [Ch] and even for finite groups by Field and Richardson IF] and [FR]. This leads to the problems of: explicitly determining the action of a group on the branches of solutions for equivariant bifurcation problems, deciding when branches with nonmaximal isotropy exist and deducing their orbit properties, as well as determining the existence of solutions with maximal isotropy when the equivariant branching lemma [C] [V] cannot be applied. Here we shall prove several results which allow one to give answers for these problems in various circumstances when the group of symmetries is finite. Because the problems for morsifications are special cases of those for bifurcations, we also give answers for them as well. 1 Partially supported by a grant from the National Science Foundation and a Fulbright Fellowship

81 To describe the results, we suppose we are given a representation of a finite group G on ~n. We consider a G-equivariant germ F(x,k): ~1 n+l,0 ---+ g:ln,0 (where G acts trivially on the last factor of ~n+l), which describes the bifurcation of a G-equivariant finite map germ f(x): ~n,0 ~ ~in,0. G acts on the set of branches F-I(0). Alternatively if h(x): ~tn,0 ----* ~t,0 is a G-equivariant germ (with trivial action on Dq) having an isolated singularity at 0, we consider H(x,k), a G-equivariant morsification of h. Then, there is an action of G on the branches of critical points of H. We will describe these actions of G via various permutation representations of G. By way of contrast, we remark that for equivariant morsifications in the complex case, Roberts JR] gives a single group character to describe the permutation representation on the branches of critical points. In moving from the complex to the real case three significant changes takeplace: 1) some branches may be purely imaginary and do not appear as real branches; 2) the branching structure for k < 0 may differ from that for k > 0; and 3) gradx(H ) need no longer preserve orientation, and, in fact, det(gradx(H)) has sign = (-1)indx (H), where indx(H ) denotes the Morse index of H(.,k) at x. We not only wish to understand the action of G on the real branches, but also on subsets of branches satisfying the conditions in 2) and 3). The above three questions apply as well to bifurcations. We give the answers for the more general case of equivariant bifurcations which includes morsifications as a special case. Corresponding to the three situations above we have either permutation representations formed by the action of G on the set of real branches or virtual permutation representations defined using sign(k) and sign(det(gradx(H)) ) for cases 2) and 3). These representations have characters: Zp the permutation character, ~b the bifurcation character, and Xd the degree character. In theorems 1 and 2 we give formulas for these characters. From these characters, we can find the characters for the permutation representations on the various subsets. The correct formulation of the results requires the use of modular characters (in characteristic 2). However, throughout this paper we shall only use the term modular character to mean the restriction of the regular complex character of a representation to the odd order elements of the group G. A modular character still contains considerable information about the representation, e.g. its value at 1G still gives the dimension of the representation. Also, in several important cases (see corollary 2) the formulas are valid for the ordinary complex characters. These characters are computed via the G-signatures of multiplication pairings on certain algebras associated to each case. We concentrate on the semi-weighted homogeneous case where the formulas take an especially simple form. Here the algebras are: 5(F), the Jacobian algebra, ~F), the bifurcation algebra, and Q(f), the local algebra of f; these algebras have "middle weights" Sp, s b, and s d and the G-signature need only be computed on the middle

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