Invariant theory for wreath product groups

Invariant theory is an important issue in equivariant bifurcation theory. D ynamical systems with wreath product symmetry arise in many areas of applie d science. In this paper we develop the invariant theory of wreath product L(sic)G where L is a compact Lie group (in some cases, a finite group) and G is a finite permutation group. For compact L we find the quadratic and cu bic equivariants of L(sic)G in terms of those of L and G. These results are sufficient for the classification of generic steady-state branches, whenev er the appropriate representation of L(sic)G is 3-determined. When L is com pact we also prove that the Molien series of L and G determine the Molien s eries of L(sic)G. Finally we obtain 'homogeneous systems of parameters' for rings of invariants and modules of equivariants of wreath products when L is finite. (C) 2000 Elsevier Science B.V. All rights reserved. MSC. 13A50, 34A47, 58F14, 20C40, 20C35.