RUI: Structure and Representations of Finite Groups

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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.