5 years ago

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

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

206 smallest codimension

206 smallest codimension of a GL(V) G orbit of elements of Hom(V,U) G of non maximal rank, and dim V G < dim U G + IX, then Dq) is a submersion. This implies that F is + ~G equivalent to a trivial extension of a G-versal unfolding of f. Thus it remains to prove that It = It(Y). For each irreducible representation V i, let ~'i = H°m(yi,Vi) G. Then ~i = R,¢ or ~ [A]. Moreover, the isotypic decompositions of V and U may be written as V= ~ ,(Vi®ki~,iP'i) and. U= I~ ,( Vi®,R. i~z irli .)' where Iti = dim~. i HOmG(yi,V)and rli = dim~ i HomG(Yi,U). Note that the fin'st hypothesis of the corollary says that Iti>_rli. Schur's lemma implies that Hom(y,u)G~Hom~i(%ilai, ki rli) and GL(Y) G m rl GL(~ti;'Izi). The smallest codimcnsion of a GL(J.ti;ki) orbit of elements of Homki(~itti,~-.i rli) of non maximal rank is IxidirnR%i = IX(Vi,V), and thus the smallest codimcnsion of a GL(Y) G orbit of elements of Hom(V,U) O of non maximal rank is Ix(V), as required.

3 FINITE DETERMINACY 207 Good determinacy estimates are an invaluable aid to the classification of singularities. In this section we describe a version of the estimates of Bruce, du Plessis and Wall [BPW]. Their approach to obtaining estimates is particularly useful when distinguished parameters and group actions are present. See [G] for a more detailed discussion in the context of equivariant bifurcation problems. G Recall that a vector subspace, M, of Epq is said to have finite codimension if the G quotient space, EPq, is finite dimensional over ~. M Definition 3.1 G G (i)If M is any vector subspace of Epq we say that F ~ Epq is M-determinedif + F + p is 9~G-equivalent to F for all p in M. G (ii)We say that F e 8pq is finitely determined if it is M-determined for some M which G has finite codimension in ~pq. The discovery of Bruce, du Plessis and Wall was that the best possible estimates (in a well defined sense) can be obtained by considering subgroups of the original group of + equivalences which are unipotent. A subgroup of 9~ G is said to be unipotent if the corresponding (algebraic) group of 1-jets, at 0, of the equivariant diffeomorphisms : W*~V,0 ~ W*~V,0 is unipotent in the usual sense. We can identify these 1-jets with equivariant linear mappings. For any unipotent snbgroup, N, of equivariant linear mappings we define :- + + ~N = { (tI'~g'O0 ~ 5~G : J 1~(0) ~ N }. We will always take our unipotent group of equivalences to be of this form. By choosing an appropriate coordinate system on W*~V, the subgroups N can always be taken to be subgroups of the group of upper triangular equivariant matrices with ls on the diagonal. See the proof of Theorem 4.5 for an example.

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