Isomorphism between systems of equivariant singularities
type: Item
instance of: scholarly article
title: Isomorphism between systems of equivariant singularities
zbMATH Open document ID: 0934.58044
author: Duo Wang
author: Yun Tang
author: Xiao-Feng Wang
published in: Journal of Mathematical Analysis and Applications
publication date: 2000-04-09T00:00:00Z
review text: 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)].
Mathematics Subject Classification ID: 58K40
zbMATH DE Number: 1218635
zbMATH Keywords: equivariant singularities
zbMATH Keywords: Lie group representation
zbMATH Keywords: smooth germs
zbMATH Keywords: universal unfolding
zbMATH Keywords: codimension
zbMATH Keywords: normal forms
zbMATH Keywords: degeneracy and bifurcations
zbMATH Keywords: perturbations
zbMATH Keywords: dihedral group
reviewed by: Aleksandr G. Aleksandrov
MaRDI profile type: Publication
full work available at URL: jmaa.1998.5996
OpenAlex ID: W2038032524
cites work: Q3027946
cites work: Classification and unfoldings of degenerate Hopf bifurcations
cites work: Singularities and groups in bifurcation theory. Volume I
cites work: Singularities and groups in bifurcation theory. Volume II
cites work: The recognition problem for equivariant singularities
cites work: Q4001520
DOI: 10.1006/JMAA.1998.5996
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description: scientific article; zbMATH DE number 1218635