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

RUI：有限群表示的伽罗瓦自同构和局部全局性质

基本信息

批准号：
2100912
负责人：
Mandi Schaeffer Fry
金额：
$ 15.44万
依托单位：
Metropolitan State University of Denver
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2021
资助国家：
美国
起止时间：
2021-09-01 至 2024-08-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=2100912&HistoricalAwards=false
关键词：
RUI Galois Automorphisms Local Global

项目摘要

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.

这个项目是在群论和有限群的表示理论领域。群可以被理解为对称性的集合，而群论的研究是出于对理解物体对称性的渴望，无论是在自然界、艺术、通信网络还是任何其他对称性可能发挥作用的地方。群论在物理、化学和其他自然科学中有应用。近年来，群论的研究对技术进步产生了重大影响，例如密码学和编码理论。表示论是用来更好地理解群的结构和它所表示的对称性的工具。表示提供了一种将抽象群视为矩阵群的方法，其结构通常更容易理解。这个项目的重点是一些问题，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组件，非常适合本科生参与，并将他们引入群论和数学研究。本研究的主要工作是招募、鼓励和指导学生进行本科研究项目。更具体地说，本研究项目所考虑的问题需要理解有限李群的不可约特征标，并涉及将群的特征标理论与其局部子群的特征标联系起来，通过一系列被称为本地-全球网络的网络。局部-整体哲学的中心思想是，有限群的表示论的关键信息可以从其局部子群的表示论的知识中推导出来。第一个局域-全局猜想，也是目前该领域问题的主要动机之一，被称为麦凯猜想。尽管进行了大量研究，但这个猜想对群论家来说仍然是个谜。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本身的特征标理论的信息，人们提出了几种更强的麦凯猜想形式，本项目考虑了涉及伽罗瓦自同构（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，该项目中的几个问题旨在进一步研究李群及其局部子群的块，以及伽罗瓦自同构对这些对象的作用。由于各种群作用对这些特征的参数化的影响是一些局部-全局拓扑和其他关于Lie型群表示的重要问题中特别有问题的组成部分，该项目的研究成果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，并通过利用基金会的智力价值进行评估，被认为值得支持和更广泛的影响审查标准。