An optimization problem for unitary and orthogonal representations of finite groups

Abstract:

Let $G \to {\text {GL}}(V)$ be a faithful orthogonal representation of a finite group G acting in an Euclidean space V. For a unit vector x we choose $g \ne 1$ in G so that $|gx - x|$ is minimal and put $\delta (x) = |gx - x|$. We study the class of vectors x which maximize $\delta (x)$ and have the additional property that $|gx - x|$ depends only on the conjugacy class of $g \in G$. For some special types of representations we are able to characterize completely this class of vectors.