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

Do absolutely irreducible group actions have odd dimensional fixed point spaces?