Pattern selection with O(3) symmetry

Abstract

For each irreducible representation of SO(3) and O(3) we determine, up to conjugacy, all isotropy subgroups and identify, in particular, the maximal isotropy subgroups. Each isotropy subgroup corresponds to a possible planform for the spherical Bénard problem. Using an equivariant branching lemma of Cicogna [1] we prove, for each of these representations, the existence of solutions corresponding to a number of different planforms, thus extending substantially the work of Busse [2, 3] and Sattinger [4]. We also give a useful criterion for showing when solutions obtained by the equivariant branching lemma must be unstable.