Integrated study of probability and representation theory towards harmonic analysis on huge groups

大群调和分析的概率与表示论综合研究

• 批准号: 19340032
• 负责人: HORA Akihito
• 金额: $ 8.99万
• 依托单位: Nagoya University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (B)
• 财政年份: 2007
• 资助国家: 日本
• 起止时间: 2007 至 2010
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-19340032/
• 关键词: 解析学; 関数解析学; 確率論; 表現論; 調和解析; 表現の指標; 対称群; ヤンググラフ; 環積; 無限対称群; 漸近挙動; 指標; 量子確率論

Towards developing harmonic analysis on huge groups, we did integrated studies of probability theory and group representations. Harmonic analysis is a discipline which seeks deep mathematical structures by looking at symmetries of the objects and develops analysis relying on them. In this study, we are led to huge groups describing the symmetries because our objects are so big as to have an infinite degree of freedom. Main results among the ones we obtained are (i) classification and explicit formulas of the characters which are building blocks of harmonic analysis, and (ii) a series of theorems which construct a bridge between asymptotic behavior of representations of groups and probabilistic limit theorems.

为了发展大型群的调和分析，我们对概率论和群表示进行了综合研究。调和分析是一门通过研究对象的对称性来寻求深层数学结构并依赖于它们进行分析的学科。在这项研究中，我们被引导到描述对称性的巨大群体中，因为我们的对象如此之大，以至于具有无限的自由度。我们得到的主要结果是（i）特征标的分类和显式公式，这些特征标是调和分析的基石;（ii）一系列定理，它们在群表示的渐近行为和概率极限定理之间架起了一座桥梁。

项目成果

Limits of characters of wreath products Sn (T) of a compact group T with the symmetric groups and characters of S∞ (T), I.

紧群T与S∞(T)、I的对称群及特征的花环积Sn(T)的特征极限。

• 发表时间: 2009
• 期刊: Nagoya Mathematical Journal 193
• 作者: T. Hirai;E. Hirai;A. Hora
• 通讯作者: A. Hora

Limits of characters of wreath products Sn(T) of a compact group T with the symmetric groups and characters of S∞(T), II. From a viewpoint of probability theory

具有对称群S∞(T)的紧群T的花圈乘积Sn(T)的性质极限，从概率论的角度看。

• 发表时间: 2008
• 期刊: Journal of Mathematical Society of Japan 60
• 作者: A. Hora;T. Hirai;E. Hirai
• 通讯作者: E. Hirai

Characters and harmonic functions related to infinite wreath product groups

与无限花环产品组相关的特征和调和函数

• 发表时间: 2010
• 作者: N. Kobayashi;H. Tokoro;Y. Hirahara;ほか6名;A.Hora
• 通讯作者: A.Hora

Limits of characters of wreath products Sn(T) of a compact group T with the symmetric groups and characters of S∞(T), II.

具有对称群S∞(T)的紧群T的花圈积Sn(T)的性质极限及其性质，II．

• 发表时间: 2008
• 期刊: From a viewpoint of probability theory, Journal of Mathematical Society of Japan Vol.60
• 作者: 3A.Hora;T.Hirai;E.Hirai
• 通讯作者: E.Hirai

Kerov's CLT for a non-Plancherel ergodic measures

非 Plancherel 遍历测度的 Kerov 的 CLT

• 发表时间: 2008
• 作者: F. Iwamuro;他21名;A.Hora
• 通讯作者: A.Hora