Equivariant Bifurcation and Absolute Irreducibility in $\R^8$ A contribution to the Ize conjecture

Abstract

M. Field [5] refers to an unpublished work by J. Ize for a result that loss of stability through an absolutely irreducible representation of a compact Lie group leads to bifurcation of steady states. The main ingredient of the proof is the hypotheses, that for an absolutely irreducible representation of a compact Lie group there exists a closed subgroup whose fixed point space is odd dimensional. Then, using Brouwer degree, one gets the result. We refer to the hypotheses that for an absolutely irreducible representation of a compact Lie group there exists at least one subgroup with an odd dimensional fixed point space as the algebraic Ize conjecture (AIC). Lauterbach and Matthews [10] have shown that the (AIC) is in general not true. In fact they have constructed three infinite families of finite subgroups of SO{4} which act absolutely irreducibly on $\R^4$ and for each of them any isotropy subgroup has an even dimensional fixed point space. Moreover in [10] it is shown that in spite of this failure of the (AIC) the original conjecture is true at least for groups in two of these three families. In this paper we show a similar bifurcation result for the third family defined in [10]. We go on and construct a family of groups acting absolutely irreducibly on $\R^8$ which have only even dimensional fixed point spaces. Then we discuss the steady state bifurcations in this case. We end this paper with a discussion on how to extend the results in [10] to larger sets of groups which act on $\R^{4}$.