Invariant theory for wreath product groups

Invariant theory is an important issue in equivariant bifurcation theory. Dynamical systems with wreath product symmetry arise in many areas of applied science. In this paper we develop the invariant theory of wreath product where is a compact Lie group (in some cases, a finite group) and is a finite permutation group. For compact we find the quadratic and cubic equivariants of in terms of those of and . These results are sufficient for the classification of generic steady-state branches, whenever the appropriate representation of is 3-determined. When is compact we also prove that the Molien series of and determine the Molien series of . Finally we obtain ‘homogeneous systems of parameters’ for rings of invariants and modules of equivariants of wreath products when is finite.