Invariant theory for wreath product groups

Abstract

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 L ≀ 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 cubic equivariants of L ≀ G in terms of those of L and G . These results are sufficient for the classification of generic steady-state branches, whenever the appropriate representation of L ≀ G is 3-determined. When L is compact we also prove that the Molien series of L and G determine the Molien series of L ≀ G . Finally we obtain ‘homogeneous systems of parameters’ for rings of invariants and modules of equivariants of wreath products when L is finite.