  In his volume [5] on "Symmetry Breaking for Compact Lie Groups" Mike Field
quotes a private communication by Jorge Ize claiming that any bifurcation
problem with absolutely irreducible group action would lead to bifurcation of
steady states. The proof should come from the fact that any absolutely
irreducible representation possesses an odd dimensional fixed point space.
  In this paper we show that there are many examples of groups which have
absolutely irreducible representations but no odd dimensional fixed point
space. This observation may be relevant also for some degree theoretic
considerations concerning equivariant bifurcation. Moreover we show that our
examples give rise to some interesting Hamiltonian dynamics and we show that
despite some complications we can go a long way towards doing explicit
computations and providing complete proofs. For some of the invariant theory
needed we will depend on some computer aided computations. The work presented
here greatly benefited from the computer algebra program GAP [6], which is an
indispensable aid for doing the required group theory computations.
