RUI: Galois Automorphisms and Local-Global Properties of Representations of Finite Groups

本课题的研究方向是群论和有限群的表示理论。群体可以被理解为对称的集合，研究群论的动机是理解物体的对称性，无论是在自然界、艺术、通信网络，还是任何其他对称可能发挥作用的地方。群论在物理、化学和其他自然科学中都有应用。近年来，群论的研究对密码学和编码理论等技术进步产生了重大影响。表征理论是一种用来更好地理解群体结构及其所代表的对称性的工具。表示提供了一种将抽象组视为一组矩阵的方法，其结构通常更容易理解。这个项目的重点是一些问题，这些问题试图将有限群的表示理论与某些所谓的局部子群的结构和表示联系起来，这些局部子群反映了由群编码的数字信息。该项目中要研究的几个问题涉及计算和其他非常适合本科生参与的组件，并向他们介绍群论和数学研究。在这个项目下，研究者活动的一个关键部分将是招募、鼓励和指导学生从事本科研究项目。更具体地说，在这个项目中考虑的问题需要理解李型有限群的不可约特征，并涉及到一个群的特征理论与其局部子群的特征之间的联系，通过一组被称为局部-全局猜想的猜想。局部-全局哲学的核心思想是，关于有限群的表示理论的关键信息可以从其局部子群的表示理论的知识中推断出来。最早的这些局部-全局猜想之一，也是目前该领域问题的主要动机之一，被称为麦凯猜想。尽管对这个猜想进行了大量的研究，但对于群论学家来说，这个猜想仍然是一个谜。为了更好地理解为什么局部子群似乎提供了如此多关于群体本身特征理论的信息，已经提出了几种更强形式的McKay猜想，本项目考虑了那些涉及伽罗瓦自同构（McKay- navarro猜想），块理论（Alperin-McKay猜想）以及两者的组合（Alperin-McKay- navarro猜想）的作用。因此，项目中的几个问题旨在进一步研究李型群的块及其局部子群，以及伽罗瓦自同构在这些对象上的作用。由于各种群体行为对这些特征的参数化的影响在许多局部-全局猜想和其他关于Lie类型群体表示的重要问题中是一个特别有问题的组成部分，因此本项目的研究将应用于该领域的其他问题。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

This project is in the area of group theory and the representation theory of finite groups. Groups may be understood as collections of symmetries and the study of group theory was motivated by the desire to understand the symmetry of an object, whether it be in nature, art, communication networks, or any other place that symmetry might play a role. Group theory has applications in physics, chemistry, and other natural sciences. In recent years, research in group theory has had a significant impact on technological advances, such as in cryptography and coding theory. Representation theory is a tool used to better understand the structure of a group and the symmetries it represents. Representations provide a way to view an abstract group as a group of matrices, whose structure is often easier to understand. This project focuses on a number of problems which seek to relate the representation theory of a finite group to the structure and representations of certain so-called local subgroups, which reflect numerical information encoded by the group. Several problems to be studied in the project involve computations and other components that are well-suited for involving undergraduate students and introducing them to group theory and mathematical research. A key part of the investigator's activities under the project will be to recruit, encourage, and mentor students to pursue undergraduate research projects.More specifically, the problems under consideration in this project require understanding the irreducible characters of finite groups of Lie type, and involve relating the character theory of a group to the characters of its local subgroups, through a collection of conjectures known as local-global conjectures. The local-global philosophy centers around the idea that critical information about the representation theory of a finite group can be deduced from knowledge of the representation theory of its local subgroups. One of the first of these local-global conjectures, and currently one of the main motivations for problems in the area, is known as the McKay conjecture. Although heavily studied, this conjecture is still somewhat of a mystery to group theorists. In pursuit of a better understanding of why local subgroups seem to provide so much information about the character theory of the group itself, several stronger forms of the McKay conjecture have been proposed, and this project considers those involving the role of Galois automorphisms (the McKay-Navarro conjecture), block theory (the Alperin-McKay conjecture), and the combination of the two (the Alperin-McKay-Navarro conjecture). Hence, several of the questions in the project aim to further the study of the blocks of groups of Lie type and their local subgroups, as well as the action of Galois automorphisms on these objects. Since the effect of various group actions on parametrizations of these characters is an especially problematic component in a number of local-global conjectures and other important problems regarding representations of groups of Lie type, the research in this project will have applications to other problems in the area.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.