Equivariant bifurcations and morsifications for finite groups
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For a bifurcation germ F(x,λ):ℝn+1,0→ℝn,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−1(0). These sets include the set of all branches as well as the set of branches where λ>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.
KeywordsVirtual Character Permutation Representation Local Algebra Degree Character Regular Character
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