Cyclic eigenvector obstruction to invariant vectors
= Cyclic eigenvector obstruction to invariant vectors
In a <unitary representation> of a <finite group>, if a cyclic vector $v$ satisfies $\rho(g)v=av$ for some $a\ne1$, then there are no invariant vectors. Indeed the averaging <orthogonal projection> $P=|G|^{-1}\sum_h\rho(h)$ satisfies $P\rho(g)=P$, so $Pv=aPv=0$. All translates of $v$ also project to zero, and they span the representation.