Representation theory of finite groups, I

This course starts from the beginning in representation theory of finite groups, with a few prerequisites in finite groups, linear algebra, and rings and fields. It develops the foundations of the subject and introduces two of the main early applications of representation theory, called Burnside's p^αq^β-theorem, and Frobenius's theorem.

This course follows a slightly less standard approach, preferring to develop a bit of abstract theory first and then use it to quickly prove the core of group representation theory, rather than the slower but more concrete approach one may find in some standard textbooks.

Chapter 1 introduces the general theory of modules, with sections on tensor products and Hom-spaces, Wedderburn's theorem, Maschke's theorem and the decomposition of the group algebra, and Schur's lemma.

Chapter 2 looks at character theory. Here we see row and column orthogonality, the Frobenius-Schur indicator, linear characters, and representations of tensor products. We end by constructing the character tables of all finite abelian groups.

Chapter 3 looks at character values, and particularly their number-theoretic properties. The highlights in this chapter are the result that character degrees divide the order of the group and Burnside's p^αq^β-theorem.

Chapter 4 introduces the last major construction in this course: induction. We prove Frobenius reciprocity, then the Mackey formula and finally Frobenius's theorem in the final lecture of the course.

There are a number of good textbooks in this area. The most commonly recommended is the following:

Gordon James and Martin Liebeck. Representations and characters of groups. 2nd ed. Cambridge University Press, 2001.

A few others are as follows.

Martin Burrow. Representation theory of finite groups. Dover, 1993.

I Martin Isaacs. Character theory of finite groups. Dover, 1994.

Walter Ledermann. Introduction to group characters. 2nd edition. Cambridge University Press, 1987.

Peter Webb. A course in finite group representation theory. Cambridge University Press, 2016.

The lecture notes are available as a single file (PDF,?? kB), and there is an html version with the lecture videos integrated.

0. 0 Preliminaries
   •   0.1 Preliminaries from groups
   •   0.2 Preliminaries from rings
   •   0.3 Preliminaries from fields
   •   0.4 Preliminaries from linear algebra
1. 1 Matrix groups, algebras and modules
   •   1.1 Some new vector spaces
   •   1.2 Irreducible matrix groups
   •   1.3 Wedderburn's theorem
   •   1.4 Algebras and modules
   •   1.5 Maschke's theorem
   •   1.6 Homs and tensors
2. 2 Character theory
   •   2.1 Basic definitions and properties
   •   2.2 The Frobenius-Schur indicator
   •   2.3 Kernels of characters
   •   2.4 1-dimensional representations of groups
   •   2.5 Direct products and abelian groups
3. 3 Algebraic integers and characters
   •   3.1 Algebraic integer preliminaries
   •   3.2 Applications of central elements
   •   3.3 Burnside's p^αq^β-theorem
4. 4 Induction and applications
   •   4.1 Induction
   •   4.2 Mackey formula
   •   4.3 Frobenius groups

There are exercises at the end of each chapter.

1. Exercises for Chapter 1
2. Exercises for Chapter 2
3. Exercises for Chapter 3
4. Exercises for Chapter 4

There are very few prerequisites in this course beyond some standard facts from linear algebra and ring theory, which are summarized in Lecture 1, or Chapter 0 of the notes.

After this course, there are a number of directions one may take: in ordinary representation theory, there is Representation theory of finite groups, II. One may shift to modular representation theory if one has taken a course in homological algebra. One can specialize to representation theory of symmetric groups.