Study on decomposition of unitary representations of groups and harmonic functions on branching graphs

我们打算在一个大的群体上发展谐波分析，它在澄清涉及大量因素的复杂现象方面发挥作用。我们的项目旨在建立一个具体的框架，通过这个框架，人们可以清楚地理解一个大群体的表征分解。对于紧群的环积，包括无限对称群和无限复反射群，我们采用了概率论、群表示和势理论的交互方法，在群特征公式及其表征、分支图和最小调和函数的边界以及图的路径空间上遍历概率的描述等方面得到了具体而详细的结果。

We intend to develop harmonic analysis on a large group, which plays a role in clarifying complex phenomena involving a huge number of factors. Our project aims at establishing a concrete framework by which one clearly understands decomposition of representations of a large group. For wreath products of a compact group, including the infinite symmetric group and infinite complex reflection groups, we took interactive approach from probability theory, group representations and potential theory, and obtained concrete and detailed results on such objects as group character formulas and their characterization, boundaries of the branching graph and minimal harmonic functions, and description of ergodic probabilities on the path space of the graph.