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

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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.