Spontaneous symmetry breaking in higher dimensional fixed point spaces (Q1194077)

The author studies the bifurcation of zeros for one-parameter families of maps \(f_ \lambda: V\to V\) which are equivariant with respect to an irreducible representation of \(SO(3)\) on \(V\). Fixing a subgroup \(\Sigma\) of \(SO(3)\) with two-dimensional fixed point space, he investigates the question whether generically a branch of solutions with isotropy \(\Sigma\) exists. The main result of the paper is concerned with the case where \(\Sigma\) is a dihedral group which is also a submaximal isotropy group. It states that the existence of a branch with isotropy \(\Sigma\) depends on two numbers in the quadratic part of \(f_ 0\). These numbers can be computed in some cases using Clebsch-Gordan coefficients.