Title of the course: An Introduction to character theory
Instructor: Dr. Thomas Keller
Institution: Texas State University
Dates: 22-28 July 2024
Prerequisites: Students should be familiar with basic linear algebra as well as group theory: group actions, Sylow theorems, etc.
Level: Graduate students or advanced undergraduate students
Abstract: Representation theory is the study of finite groups by representing the group elements as linear transformations of vector spaces; in other words, every group element is represented by an invertible matrix over a field. The case that this field is the field of complex numbers is particularly important and will be the focus of this course. In this case, if instead of the matrices one just studies their traces (i.e., the sums of the main diagonal entries), one can retain a lot of the information about the representation (and thus the group), and of course the trace is just a complex number and is thus much easier to study than a large matrix. The function assigning to the group elements the traces of a given representation is called a character. We will study character theory of finite groups in this course.
It turns out that finite groups have only finitely many characters, and that many important results in finite group theory can only be proved or have much easier proofs using character theory.  A classical example is a theorem of Burnside stating that groups which have order divisible by at most two distinct primes is necessarily solvable. We will introduce characters and their basic properties (e.g. orthogonality relations) and as time permits, provide a proof of Burnside’s result.
Language: EN