RUI: Structure and Representations of Finite Groups

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对象的对称性，无论是在自然界，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、化学和其他自然科学中有应用，近年来，群论和其他代数领域的研究也对技术进步产生了重大影响，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。粗略地说，表示提供了一种将抽象群视为矩阵集合的方法，矩阵的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群的结构联系起来，这反过来可能会让人们更深入地了解真实世界的对象，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产生影响。在考虑的几个问题涉及计算和其他组件，非常适合涉及本科生，并介绍他们的群论和数学研究，研究者将招募，鼓励，并指导学生从事与此项目有关的研究活动。更具体地说，这个项目是集中在李型有限群的不可约字符，它们构成了有限单群的最大集合。 它涉及有关的字符理论的一组，某些小组，通过当地的全球aerotures和不可约字符的限制，除了有关字符领域的价值的性质共轭类的组。该项目中的几个问题旨在进一步了解该领域中关于伽罗瓦自同构对李群特征的作用的现有知识。 由于各种群作用对这些特征的参数化的影响是一些局部-全局解和其他关于李群表示的主要问题中特别有问题的组成部分，这对该领域的许多其他问题都很有意义。特别是，该项目的一部分涉及李型群的Navarro Galois-McKay猜想，以及研究李型群的共轭类和特征的现实。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。

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.