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RUI: Structure and Representations of Finite Groups

RUI：有限群的结构和表示

基本信息

批准号：
1801156
负责人：
Mandi Schaeffer Fry
金额：
$ 11万
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：
2018
资助国家：
美国
起止时间：
2018-08-01 至 2022-07-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1801156&HistoricalAwards=false
关键词：
RUI Structure Representations Finite Groups

项目摘要

This project is in the area of group theory and the representation theory of finite groups. The study of group theory is 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. Finite group representation theory has applications in physics, chemistry, and other natural sciences, and in recent years, research in group theory and other algebraic areas has also 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. Roughly speaking, representations provide a way to view an abstract group as a collection of matrices whose structure is often easier to understand. This project focuses on a number of problems that seek to relate the representation theory of a finite group to the structure of the group, which in turn may give more insight into the real-world objects whose symmetries are encoded in these groups and have implications for the various applications of group theory. Several of the problems under consideration involve computations and other components that are well-suited for involving undergraduate students and introducing them to group theory and mathematical research, and the investigator will recruit, encourage, and mentor students to pursuing research activities related to this project.More specifically, this project is focused on irreducible characters of finite groups of Lie type, which make up the largest collection of finite simple groups. It involves relating the character theory of a group to that of certain subgroups, through local-global conjectures and irreducible character restrictions, in addition to relating character fields of values to properties of conjugacy classes of the group. Several of the questions in the project aim to further the current knowledge in the field regarding the action of Galois automorphisms on characters of groups of Lie type. 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 main problems regarding the representations of groups of Lie type, this is of interest to many other problems in the area. In particular, part of the project concerns Navarro's Galois-McKay conjecture for groups of Lie type, as well as studying reality for conjugacy classes and characters of groups of Lie type.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。研究群论的动机是希望了解一个对象的对称性，无论它是在自然界、艺术、通信网络或任何其他地方，对称性可能发挥作用。有限群表示理论在物理、化学等自然科学中都有应用，近年来，群论和其他代数领域的研究也对密码学和编码学等技术的进步产生了重大影响。表象理论是用来更好地理解群的结构和它所代表的对称性的工具。粗略地说，表示法提供了一种将抽象组视为矩阵集合的方法，这些矩阵的结构通常更容易理解。这个项目集中在一些问题上，这些问题试图将有限群的表示理论与群的结构联系起来，这反过来可能使我们更深入地了解现实世界中的对象，这些对象的对称性被编码在这些群中，并对群论的各种应用具有影响。一些正在考虑的问题涉及计算和其他组件，非常适合让本科生参与并将他们介绍给群论和数学研究，调查人员将招募、鼓励和指导学生从事与此项目相关的研究活动。更具体地说，此项目专注于有限李型群的不可约特征标，它构成了有限单群的最大集合。它涉及到通过局部-整体猜想和不可约特征标限制将群的特征标理论与某些子群的特征标理论联系起来，以及将值的特征标域与群的共轭类的性质联系起来。该项目中的几个问题旨在加深该领域中关于Galois自同构在Lie类型群的特征标上的作用的现有知识。由于各种群作用对这些特征的参数化的影响在许多局部-全局猜想和关于Lie类型群的表示的其他主要问题中是一个特别有问题的组成部分，这对该领域的许多其他问题是感兴趣的。特别是，该项目的一部分涉及纳瓦罗关于Lie类型群的Galois-McKay猜想，以及研究共轭类和Lie类型群的特征的实在性。该奖项反映了NSF的法定使命，并通过使用基金会的智力价值和更广泛的影响审查标准进行评估，被认为值得支持。