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

TL;DR

This paper establishes bounds on the width of transitive sets on the sphere and on matrix coefficients of finite groups, revealing limitations on how such sets and group actions can be structured in high dimensions.

Contribution

It provides the first bounds on the width of transitive sets and matrix coefficients of finite groups, answering a question posed by Yufei Zhao.

Findings

The width of any transitive set in  is bounded by a constant times ()^{-1/2}.

For any finite group representation, there exists a vector with small maximum inner product under the group action.

The diameter of the quotient of the sphere by a finite group approaches cb5 as dimension increases.

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 $(\log d)^{-1/2}$. This is a consequence of the following result: If $G$ is a finite group and $\rho : G \rightarrow \mbox{U}_d(\mathbf{C})$ a unitary representation, and if $v \in \mathbf{C}^d$ is a unit vector, there is another unit vector $w \in \mathbf{C}^d$ such that \[ \sup_{g \in G} |\langle \rho(g) v, w \rangle| \leq (1 + c \log d)^{-1/2}.\] These results answer a question of Yufei Zhao. An immediate consequence of our result is that the diameter of any quotient $S(\mathbf{R}^d)/G$ of the unit sphere by a finite group $G$ of isometries is at least $\pi/2 - o_{d \rightarrow \infty}(1)$.