Invariant theory for wreath product groups
Invariant theory is an important issue in equivariant bifurcation theory. In the paper the authors develop the invariant theory of wreath products of a compact Lie group with a finite permutation group. For a compact Lie group the authors find the quadratic and cubic equivariants of the wreath product in terms of those of the Lie group and the permutation group. As one of the technical tools the authors prove that the Molien series of the wreath product is determined by the Molien series of the factors. Moreover they obtain homogeneous systems of parameters for rings of invariants and modules of equivariants of wreath products when the Lie group is supposed to be finite. In essence they prove that useful information about the invariants and equivariants of the wreath product can be read off in a simple manner from similar information about the factors. To this end they modify algorithms of \textit{B. Sturmfels} [``Algorithms in invariant theory'', Texts and Monographs in Symbolic Computation (Wien 1993; Zbl 0802.13002)], and of \textit{P. A. Worfolk} [J. Symb. Comput. 17, 487-511 (1994; Zbl 0819.13013)] for the Molien series and Cohen-Macaulay property to the wreath product.
scientific article; zbMATH DE number 1488534
scientific article