Workshop on Representation Theory of Finite Groups

Representation theory is a branch of mathematics that studies abstract algebraic structures by representing their elements as linear transformations of vector spaces. In essence, a representation makes an abstract algebraic object more concrete by describing its elements using matrices and their operations (matrix addition and multiplication). The theory of matrices and linear operators is well-understood, so representations of more abstract objects in terms of familiar linear algebra objects helps us in understanding more of these abstract theories.

We will mostly be learning from the book by Steinberg (see resources below). To quote from the book's introduction:

"The original purpose of representation theory was to serve as a powerful tool for obtaining information about finite groups via the methods of linear algebra, e.g., eigenvalues, inner product spaces, and diagonalization. The first major triumph of representation theory was Burnside’s pq-theorem. It was not until much later that purely group theoretic proofs were found for it. Representation theory went on to play an indispensable role in the classification of finite simple groups.

However, representation theory is much more than just a means to study the structure of finite groups. It is also a fundamental tool with applications to many areas of mathematics and statistics; both pure and applied. Fourier analysis on finite groups also plays an important role in probability and statistics, especially in the study of random walks on groups, such as card shuffling and diffusion processes, and in the analysis of data."

Resources

• [Ar] Michael Artin, Algebra, 2nd Ed. Pearson, 2010.

• [St] Benjamin Steinberg, Representation Theory of Finite Groups: an Introductory Approach. Springer, 2011.

Link: https://users.metu.edu.tr/sozkap/513-2013/Steinberg.pdf 

Notes and exercises will be posted on the website during the workshop.

The book by Steinberg will be our main reference for this series, whereas the first is exclusively for prerequisites. For information on how to get either book, you can email one of the organizers (contact emails below).

Prerequisites

We will assume a good command of basic linear algebra (vector spaces and subspaces, linear transformations, kernels and images, the rank-nullity theorem, etc.) as well as being comfortable with the definitions and basic examples of group theory (groups, homomorphisms, the first isomorphism theorem). Some important examples of groups that will play a major role in our series are the general linear group GL(V), and various finite groups. The concept of rings will show up occasionally but no knowledge of ring theory is really required.

The first four chapters of [Ar] (or equivalent content from a different book) should be very sufficient to prepare you. 

Also, here are lecture notes on linear algebra and modern algebra.

What should I do if I know these things already and want to be more prepared?

Read the first chapter of [St] and do all the exercises! That is the best way to be well-prepared. 

Structure of the Workshop