Symmetry-breaking bifurcations of wreath product systems
The structure of matrices is investigated which are commuting with special linear representations of special finite groups. This research is motivated by the bifurcation analysis of dynamical systems describing coupled oscillators. Here a coupling is investigated which leads to a dynamical system which is equivariant with respect to a linear representation of a wreath product group. In the bifurcation analysis matrices commuting with a wreath product arise naturally. This paper uses the classification of bifurcation subgroups for absolutely irreducible representations of wreath products and of \(C\)-axial subgroups of complex irreducible representations of wreath products given by \textit{B. Dionne, M. Golubitsky} and \textit{I. N. Stewart} [Nonlinearity 9, No. 2, 559-574 (1996; Zbl 0894.58049)]. The two main results of this paper are based on Schur's lemma. The authors investigate the special form of matrices (a) commuting with a representation of an axial subgroup of a wreath product group which is derived by restricting an absolutely irreducible representation of the wreath product and (b) commuting with a linear representation of a \(C\)-axial subgroup of a wreath product derived by restricting a complex irreducible representation of the wreath product. The equivariant bifurcation lemma and the equivariant Hopf bifurcation are recalled for this context.
linear representation theory of wreath product groups
scientific article; zbMATH DE number 1385272