Michael Field
Symmetry Breaking for Compact Lie Groups
Memoirs of the American Mathematical Society, 575, 1996.
This work comprises a general study of symmetry breaking for
compact Lie groups in the context of equivariant bifurcation
theory. We start by extending the theory developed by Field
and Richardson [92] for absolutely irreducible representations
of finite groups to general irreducible representations of
compact Lie groups. In particular, we allow for branches of
relative equilibria and phenomena such as the Hopf bifurcation.
We also present a general theory of determinacy for irreducible
Lie group actions along the lines previously described in Field [89].
In the main result of this work, we show that branching patterns
for generic equivariant bifurcation problems defined on irreducible
representations persist under perturbations by sufficiently
high order non-equivariant terms. We give applications of this result
to normal form computations yielding, for example, equivariant
Hopf bifurcations and show how normal form computations of
branching and stabilities are valid when we take account of the
non-normalized tail.