Isomorphism between systems of equivariant singularities
Let \(V\) be a finitely dimensional \(\mathbb R\)-vector space, let \(GL(V)\) be the set of all invertible linear transformations on \(V,\) and let \(\rho : \Gamma \rightarrow GL(V)\) be a representation of a Lie group \(\Gamma\) on \(V.\) Then \(\Gamma\) acts on \(V\) by the following rule: \(\gamma \cdot x = \rho(\gamma)x\), \(\gamma \in \Gamma\), \(x \in V.\) Denote by \({\mathcal E}_{V,k}(\Gamma)\) the set of all \(C^\infty\)-smooth \(\Gamma\)-equivariant germs \(g : (V \times {\mathbb R}^k, 0) \rightarrow V,\) where \(g(\gamma \cdot x, \mu) := \gamma \cdot g(x, \mu)\) for any \(x \in V\), \(\mu \in {\mathbb R}^k\), \(\gamma \in \Gamma.\) By definition, the set \({\mathcal L}_\rho = \prod_{k=0}^\infty {\mathcal E}_{V, 1+k}(\Gamma)\) is called a system of equivariant singularities with respect to the Lie group representation \(\rho.\) The authors obtain a sufficient condition under which two systems of equivariant singularities are isomorphic. As an application the following two concrete isomorphisms are constructed. The first one is an isomorphism between the system of \(O(n)\)-equivariant singularities in its irreducible representation on \({\mathbb R}^n\) and the system of one-dimensional \({\mathbb Z}_2\)-equivariant singularities. The second is an isomorphism between the system of equivariant singularities with respect to the \(n(n+1)/2\)-dimensional action of \(O(n)\) on symmetric \(n \times n\)-matrices and the system of \((n-1)\)-dimensional \(S_n\)-equivariant singularities. In conclusion it is remarked that the latter result for \(n=3\) was obtained by \textit{M. Golubitsky}, \textit{Ian Stewart} and \textit{David G. Schaeffer} [`Singularities and groups in bifurcation theory. II', Springer-Verlag, New York (1985; Zbl 0691.58003)].
scientific article; zbMATH DE number 1218635
scientific article