Study of asymptotic theory for representations of symmetric groups from the viewpoint of scaling limits for probability models

从概率模型标度极限的角度研究对称群表示的渐近理论

基本信息

批准号：
16540154
负责人：
HORA Akihito
金额：
$ 2.3万
依托单位：
Okayama University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
2004
资助国家：
日本
起止时间：
2004 至 2006
项目状态：
已结题

项目摘要

The main purpose of the present research is to study asymptotic behavior of various characteristic quantities of representations of symmetric groups and other similar discrete groups as the sizes of the groups grow, and to investigate the limiting pictures from the viewpoint of scaling limits in probability theory and statistical mechanics. Features to be noted in this research include using methods of limit theorems in quantum probability and making much of relations to free probability and random matrices. The following are several concrete results.1. We studied the spectral distributions of Laplacians with respect to the Gibbs states in zero temperature and infinite volume limit as graphs grow with their degrees and temperatures keeping certain scaling balances. We computed the asymptotic behavior in details under the formulation of quantum central limit theorem by using creation and annihilation operators on interacting Fock spaces.2. Through combinatorial hard analysis of moments of the Jucys-Murphy element, we studied universal understanding of concentration phenomena in various statistical ensembles consisting of Young diagrams, including those which come from irreducible decomposition of a representation of the symmetric group such as the Littlewood-Richardson coefficients. Many of them are closely related to some properties of random walks on a certain modified Young graph. Here also we applied methods of quantum probability effectively.3. Under cooperation with T. Hirai and E. Hirai, we constructed a nice factor representation which expresses any character of a wreath product of a compact group with the infinite symmetric group as its matrix element. This representation reflects directly the characterizing parameters for the character beyond a general representation of Gelfand-Raikov.

本研究的主要目的是研究随着群体的大小的增长，对称群体和其他类似离散组的各种特征数量表示的渐近行为，并从概率理论和统计机制中的缩放限制限制的角度研究限制图片。这项研究中要注意的特征包括使用量子概率的极限定理方法，并与自由概率和随机矩阵建立了很大的关系。以下是几个混凝土结果1。我们研究了laplacians在零温度和无限体积限制下相对于吉布斯状态的光谱分布，随着图形的增长，其程度和温度可以保持一定的缩放平衡。我们通过使用创建和an灭算子在相互作用的fock空间中使用量子中心限制定理的详细信息计算了渐近行为。2。通过对Jucys-Murphy元素的矩分析的结合分析，我们研究了各种统计合奏中浓度现象的普遍理解，这些统计合奏包括由年轻图组成的各种统计合奏，包括来自对对称群体（例如Littlewood-Richardson系数）的不可约解分解。其中许多与某些修改后的年轻图上的随机步行的某些特性密切相关。在这里，我们也有效地应用了量子概率的方法3。在与T. Hirai和E. Hirai的合作下，我们构建了一个不错的因素表示，该因素表示与无限对称组的紧凑型组的花环产物的任何特征作为其基质元素。该表示直接反映了gelfand-raikov一般表示超出字符的表征参数。