Nonlinearity 7 (1994) 1385-1393
We answer some longstanding questions concerning absolutely irreducible representations of compact Lie groups. Such representations provide the natural setting for steady-state equivariant bifurcation theory, see Golubitsky, Stewart and Schaeffer. We prove the existence of maximal isotropy subgroups for which there are no branches of equilibria or relative equilibria. Also, we obtain examples of complex and quaternionic maximal isotropy subgroups. A consequence of this is the existence of primary branches of nontrivial relative equilibria (rotating waves).