5 years ago

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

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

Lemma 1.1: Xhp = 2"Xp,

Lemma 1.1: Xhp = 2"Xp, Xhb = 2"Xb, and Xhd = 2"Xd 86 The proof is an immediate consequence of lemma 4.5 of [D4]. Our goal is first to give formulas for the modular characters Xp, Xb, and Xd in terms of the algebras 3~), ~), and Q(f). Second, we shall show in certain important cases that we can recover Xp~;, Xb¢, and Xd~- Third, we shall deduce from these formulas information about the number of G-orbits, branches, etc. Finally, we give as consequences of these formulas (see §3) sufficient conditions for the existence of submaximal orbits and orbits with particularisotropy. G-signature The formulas for computing the modular characters will be based on the G-signature of the multiplication pairings on the algebras. The definition of the G-signature is based on the following proposition given in [ D4]. Proposition 1.2: Given a representation of G on a finite dimensionM real vector space V and a symmetric bilinear form ¢p on V which is G-invariang i.e. cp(g.v,g-w) = cp(v,w) a/1 v,w ~ V and g ~ G, then there is a decomposition into G-invariant subspaces V = VO ~ V+ ~ V_ where V 0 is the kernel of 9, i.e. V 0 = {v ~ V : ep(v, w) = 0 for all w E V}, and ep is positive definite on V+ and negative definite on V_. Furthermore, V 0 , V+, and V_ are unique up to isomorphism as G-representations. Thus, we define Definition 1.3: the G-signature of 9 ,denoted sigG(9), = Z+ - X_ ,where X-a: denote the modular characters for the representations V+. If q~ is fixed in the discussion we may also denote it by sigG(V). If we let X-a: denote the characters for the complex representation, then we have the regular G-signature denoted by sig~:G(9). We are ready to state the formulas for the various characters. §2 Formulas for the Characters in the Semi-weighted Homogeneous Case We consider a smooth G-equivariant germ F(x,~.) : ~ln+l,0 ~ ~qn ,0 which is a bifurcation of an equivariant finite map germ f(x) : [In,0 ----, A n ,0 and has an (algebraically)

87 isolated singularity at 0. We say that such a germ F is weighted homogeneous if we may assign weights wt(xi) = a i and wt(k) --- b so that G preserves weights and each of the coordinate functions F i of F are weighted homogeneous. We denote their weights by wt(F i) = d i. Second, we say that F is semi-weighted homogeneous if (with respect to weights wt(x i) = a i) the initial part of F, in(F) = F 0, defined by the lowest weight terms in each coordinate, is weighted homogeneous in the previous sense so that F 0 defines an (algebraically) isolated singularity and the initial part of f, in(f) --- f0, defines a finite map germ. If F 0 = (F01 ..... F0n), we again let d i = wt(F0i). For imperfect bifurcation equivalence we slightly reffme this notion by saying that F is semi-weighted homogeneous for (equivafiant imperfecO bifurcation equivalence if the initial part F 0 has finite codimension for that notion of equivalence. If F is semi-weighted homogeneous, then we can assign weights to the elements of the finite dimensional algebras .~'F0), B(F0), and Q(f0)- Each of these algebras has a multiplication pairing to the top weight nonzero element in the algebra; furthermore these pairings are G- equivariant in the sense of § 1 (this follows for J(F 0) and Q(f0) from ID4], for B(F 0) it will follow from the proof given in §7). By the preceding we mean that for each such algebra A, there is a linear functional 9: A ~ ~1 which vanishes on terms of weight less than the maximal nonzero weight such that the composition of 9 with multiplication give the bilinear pairing on A. A x A ----~ A ---~ ~1 Because weights are additive for the multiplication, there is a middle weight part of the algebra which is paired with itself; while the elements of weight above the middle weight are paired with those below. Because of this, the G-signature of the pairing is determined by its restriction to the middle weight part. To describe the middle weight parts of these algebras, we let i--1 j=l Then, the middle weights for the algebras are given by Table 2.1. To denote the middle weight parts of these algebras we use the notation that the weight m part of the algebra A will be denoted by A m. If m is not an integer, then A m = (0). Table 2.1 ALGEBRA MIDDLE WEIGHT Jacobian algebra .~F0) Sp --- s - wt(~.) Bifurcation algebra q~0) s b = s - (1/2)wt(~.) Local algebra Q(f0) Sd = (1/2)s

Reading grade 6 2.A.5.b - mdk12
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