An Algebraic Criterion for Symmetric Hopf Bifurcation

The equivariant Hopf bifurcation theorem states that bifurcating branches of periodic solutions with certain symmetries exist when the fixed-point subspace of that subgroup of symmetries is two dimensional. We show that there is a group-theoretic restriction on the subgroup of symmetries in order for that subgroup to have a two-dimensional fixed-point subspace in any representation. We illustrate this technique for all irreducible representations of SO(3) on the space V_l of spherical harmonics for l even.