On the width of transitive sets: Bounds on matrix coefficients of finite groups
On the width of transitive sets: Bounds on matrix coefficients of finite groups
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DOI：
10.1215/00127094-2019-0074
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发表时间：
2018-02
影响因子：
2.5
通讯作者：
B. Green
中科院分区：
文献类型：
--
作者：
B. Green
我们说，如果有一组异构体在其上作用，则$ \ mathbf {r}^d $中的单位球的有限子集是传递的。我们表明，任何传递集的宽度在上面的恒定时间$（\ log d）^{ - 1/2} $限制。这是以下结果的结果：如果$ g $是有限的组，而$ \ rho：g \ rightarrow \ mbox {u} _d（\ mathbf {c}）$ n unitary表示，如果$ v \ in \ in \ in \ in \ in \ in \ in Mathbf {C}^d $是一个单位向量，还有另一个单位向量$ w \ in \ mathbf {c}^d $，这样\ [ \ sup_ {g \ in G} | \ langle \ rho（g）v，w \ rangle | \ leq（1 + c \ log d）^{ - 1/2}。\]这些结果回答了Yufei Zhao的问题。
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.