Fixed points in absolutely irreducible real representations

Abstract

It has been an open question whether any bifurcation problem with absolutely irreducible group action would lead to bifurcation of steady states. A positive proposal is also known as the “Ize-conjecture”. Algebraically speaking, this is to ask whether every absolutely irreducible real representation has an odd dimensional fixed point subspace corresponding to some subgroups. Recently, Reiner Lauterbach and Paul Matthews have found counter examples to this conjecture and interestingly, all of the representations are of dimension $4k$, for $k\in\mathbb{N}$. A natural question arises: what about the case $4k+2$?

In this paper, we give a partial answer to this question and prove that in any $6$-dimensional absolutely irreducible real representation of a finite solvable group, there exists an odd dimensional fixed point subspace with respect to subgroups.