On the width of transitive sets: bounds on matrix coefficients of finite groups
| Autor: | Ben Green |
|---|---|
| Rok vydání: | 2018 |
| Předmět: |
Unit sphere
General Mathematics transitive sets Transitive set Group Theory (math.GR) 01 natural sciences Combinatorics Matrix (mathematics) Mathematics - Metric Geometry Unit vector 0103 physical sciences FOS: Mathematics Mathematics - Combinatorics 0101 mathematics primitive representation Quotient Mathematics Finite group Group (mathematics) 010102 general mathematics 20D06 Metric Geometry (math.MG) generalized Fitting subgroup Unitary representation 51F25 Jordan theorem 010307 mathematical physics Combinatorics (math.CO) Mathematics - Group Theory |
| Zdroj: | Duke Math. J. 169, no. 3 (2020), 551-578 |
| DOI: | 10.48550/arxiv.1802.01904 |
| Popis: | 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 $\pi/2 - o_{d \rightarrow \infty}(1)$. Comment: 35 pages, corrected two significant errors drawn to my attention by Ashwin Sah, Mehtaab Sawhney and Yufei Zhao |
| Databáze: | OpenAIRE |
| Externí odkaz: |
|