On the width of transitive sets: Bounds on matrix coefficients of finite groups

Abstract

We say that a finite subset of the unit sphere in ${\mathbf{R}}^d$ is transitive if there is a group of isometries which acts transitively on it. We show that the width of any transitive set is bounded above by a constant times $(logd{)}^{-1/2}$.

This is a consequence of the following result: if $G$ is a finite group and $\rho :G\to U_d\left(\mathbf{C}\right)$ a unitary representation, and if $v\in {\mathbf{C}}^d$ is a unit vector, then there is another unit vector $w\in {\mathbf{C}}^d$ such that $${sup\hspace{0.17em}}_{g\in G}|\langle \rho (g)v,w\rangle |\le (1+clogd{)}^{-1/2}.$$

These results answer a question of Yufei Zhao. An immediate consequence of our result is that the diameter of any quotient $S\left({\mathbf{R}}^d\right)/G$ of the unit sphere by a finite group $G$ of isometries is at least $\pi /2-o_{d\to \infty }\left(1\right)$.