Equivariant Bifurcation and Absolute Irreducibility in $\mathbb {R}^8$: A Contribution to Ize Conjecture and Related Bifurcations

Abstract

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 Ize conjecture (IC). If the IC is true, then it follows that loss of stability through an absolutely irreducible representation of a compact Lie group leads to bifurcation of steady states. Lauterbach and Matthews have shown that the (IC) is in general not true and have constructed three infinite families of finite subgroups of $\mathop {\mathbf{SO}(4)}$ which act absolutely irreducibly on $\mathbb {R}^4$ and have no odd dimensional fixed point space. They also have shown that in spite of this failure of the (IC) the nontrivial isotropy types are generically symmetry breaking at least for the groups in two of these three families. In this paper we show a similar bifurcation result for the third family defined by Lauterbach and Matthews. We go on and construct a family of groups acting absolutely irreducibly on $\mathbb {R}^8$ which have only even dimensional fixed point spaces. Then we discuss the steady state bifurcations in this case. Key ingredients are an abstract group theoretic construction and a kind of inductive step reducing the issue of bifurcations to a problem in $\mathbb {R}^4$. We end this paper with a discussion on how to extend the results by Lauterbach and Matthews to larger sets of groups which act on $\mathbb {R}^{4}$ and $\mathbb {R}^8$. In this context we point out, that the inductive step, which is important for our arguments, does not work in general and this gives rise to interesting new questions.