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

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

84 work on bifurcation

84 work on bifurcation theory. In trying to define the analogue of a Jacobian algebra for bifurcation theory one obtains exactly this algebra. In particular, the ideal B(F) contains ~..J(F) (once one realizes this must be true, it can be readily demonstrated by expanding determinants using the Euler relation). It is well known that F-l(0) has an algebraically isolated singularity at 0 if and only if dimlRY(F ) < oo, and f is a finite map germ if and only if dimlRQ(f) < oo. Also, these two conditions imply that dim~) < 0o. Hence, our assumptions insure that all of these algebras are finite dimensional Permutation representations on branches and half-branches We wish to describe the permutation representation of G on the branches of F-l(0). There is some ambiguity in what exactly is meant by a branch of a real curve. We will adopt the convention that a real curve consists of its algebraically defined branches, each of which consist of two half branches. For example, x2-y 3 --- 0 consists of one branch but two half branches. The action of G permutes the branches of F-l(0) as well as the half branches. Denote the set of branches by B and the set of half-branches by B'. We will define the permutation representations on B and B' as well as other representations on certain subsets which contain additionalinformation. Let Vp denote the vector space with basis {e b : b ~ B}. An action of G on Vp is defined by g.e b = eg.b for b ~ B and g ~ G. Likewise we may define the permutation representation of G on B', which we denote by Vhp. Corresponding to these representations are the group characters Hp and Xhp. Second, suppose that by some method we have attached signs e(b) to the half-branches b ~ B' in such a way that e(b) = e(g.b) for all b ~ B' and g ~ G. Then, we may define virtual representations on the branches and half branches. For branches, we decompose B = B+ t.) B_ u B m , where B+, respectively B_, respectively B m consists of branches for which both half-branches have positive signs, respectively negative signs, respectively one of each sign. G preserves each of these sets; hence, we have permutation representations Ve+, V e- , and Vem with characters X + , He-, Xe m. We define the virtual representation V e = V + - V e- and virtual character Xe = Xe + - He-. Note that this definition explicitly disregards B m. For half-branches, we decompose B' = B'+ u B'_. Associated to B'-I- are permutation representations Vhe :t: with characters Xhe :t: and the virtual representation Vhe = Vhe+ - Vhe- and virtual character Hhe = Hhe +- Xhe-. While the half-branch characters are more natural to use for the actual investigation of singularities, the branch characters are easier to work with in obtaining formulas. Because

85 certain elements of G may interchange certain pairs of half-branches of a branch, we cannot always expect to have an elementary formula relating these characters. However, we will shortly give a very simple relation between these characters in an important general situation. Before doing this, we give two important special cases of the above construction. 1) bifurcation characters: for b ~ B', we assign e(b) = sign(~.) on b. Since there are no branches in ~n x {0} , this is well-defined and clearly invariant under the G-action. We denote the virtual characters and virtual representations by V b , Vhb, Xb, and Xhb. 2) degree characters: for b ~ B', we define e(b) = sign(det(dxF)) on b. Because F defines a curve with an isolated singularity at 0, det(dxF) # 0 on any half-branch. We denote these virtual characters and representations by V d , Vhd, Xd, and Xhd- Remark: We may think off as defining a parametrized family F~.(x) : U ----} ~n, where U is a neighborhood of 0. If we choose ~. > 0 and k' < 0 sufficiently small, then there is a G- equivariant bijection between B' and F~-I(0) u Fk,-l(0). Hence, we may think of the half- branch characters as defining (virtual) permutation characters for F~-I(0) u Fk,-I(0). Summarizing the constructions, we have three (virtual) characters for representations on branches: Xp the permutation character, Xb the bifurcation character, and Zd the degree character, and three corresponding characters for representations on half-branches : Xhp, Xhb , and )Chd- Also these half-branch characters can be thought of as describing virtual representations on F~-I(0) u F)j-I(0) for k > 0 and ~.' < 0 sufficiently small. Modular Characters To properly describe the general results on the various permutation representations one should use modular representations in characteristic 2. However, for the purposes of this paper, if X is the ordinary character of a complex representation then the associated modular character is the restriction XI G*, where G* denotes the set of odd order elements of G. These characters have many properties similar to those of regular characters. In the extreme case of 2-groups, the only information they carry is the dimension of the representation. Otherwise, we shall see that they contain a considerable amount of information. (A reader interested in more information about the theory of modular representations is referred to e.g. [DP] or IS]; also, properties for the characteristic 2 case are briefly summarized in [D4] ). In all that follows, we will let the preceding characters Xp, etc. denote modular characters; and we denote the corresponding complex representations and characters by adding the subscript C, e.g. VpC, Xb¢, etc. First, for modular characters there is a very simple relation between the representations on branches and half-branches.

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