CAREER: Local-Global Properties in the Representations of Finite Groups

This project is in the area of group theory and the representation theory of finite groups. Groups may be understood as collections of symmetries, whose study 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 by providing 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. The award will also support educational activities centering graduate student mentorship and undergraduate research.

These activities include local workshops for Colorado-area graduate students and postdocs and a Directed Reading Program. The latter provides a "research-like" experience for undergraduates, in which undergraduate mentees and graduate mentors work under the guidance of the PI to study a topic outside of the students’ normal coursework. The activities in the grant also include supervising undergraduate researchers, Ph.D. students, and postdocs, who will work with the PI on problems within the scope of the project.

The problems of primary interest in this project are related to variations and consequences of the McKay–Navarro (MN) and Brauer’s Height Zero (BHZ) Conjectures. The PI and collaborators have recently proved the case of the prime 2 of the MN Conjecture, and many of the goals in this proposal are also aimed at establishing strategies for attacking the case of odd primes.

In the case of BHZ, which was one of the longest-standing local-global conjectures, the PI and coauthors completed the proof of the conjecture in its entirety, and have been working toward stronger forms of the conjecture. The problems will involve the study of blocks of groups of Lie type, the action of Galois automorphisms on characters of groups of Lie type, and the interaction between the two.

Since the effect of various group actions on parameterizations of 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.

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.