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对称群和其他类似离散群表示的各种特征量随着群规模增长的渐近行为，并从概率论和尺度极限的角度研究极限图。统计力学。这项研究值得注意的特点包括使用量子概率中的极限定理方法，并充分考虑与自由概率和随机矩阵的关系。具体结果如下： 1．我们研究了拉普拉斯算子在零温度和无限体积极限下相对于吉布斯状态的谱分布，因为图随着其度数和温度的增长而增长，并保持一定的比例平衡。我们利用相互作用的Fock空间上的创生算子和湮灭算子，详细计算了量子中心极限定理下的渐近行为。 2．通过对 Jucys-Murphy 元素矩的组合硬分析，我们研究了对由杨图组成的各种统计系综中浓度现象的普遍理解，包括来自对称群表示的不可约分解的统计系综，例如 Littlewood-Richardson 系数。其中许多与某个修改后的杨图上随机游走的某些属性密切相关。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，我们构建了一个很好的因子表示形式，它可以表达以无限对称群为矩阵元素的紧群的花环乘积的任何特征。这种表示直接反映了超出 Gelfand-Raikov 一般表示的角色特征参数。