Representation theory

In this course we will cover the representation theory of finite groups. The course will begin by covering the theory in characteristic zero, including Maschke's theorem and character theory. Then we will spend the remaining time discussing modular representation theory, representations of p-groups in characteristic p.

Time + place: 

• Tue 12-14, M104

• Wed 12-14, M103

• Exercise class: Fri 12-14, M102

Contents:

1. Representation theory of an algebra

   1. Schur's lemma

   2. Simple and irreducible representations

   3. The Jordan-Hölder and Krull-Schmidt theorems

   4. Semisimple representations and the Artin-Webberburn theorem

   5. Examples

2. Representation theory of a finite group (away from the order)

   1. Maschke's theorem

   2. Orthogonality of characters

   3. Frobenius reciprocity

   4. Artin's theorem on virtual representations

   5. Examples

3. Applications

   1. The Frobenius-Schur indicator

   2. Frobenius determinants

   3. Frobenius divisibility

   4. Burnside's theorem

4. Representation theory of a finite group (at the order)

   1. Idempotents and projective covers

   2. The Cartan matrix

   3. The decomposition map

   4. Brauer characters

The precise contents will inevitably change as the semester progresses. My goal is to do every topic we cover justice, instead of trying to race to some end.

References:

• Etingof et al. - Introduction to representation theory

• Webb - A Course in Finite Group Representation Theory

• Serre - Linear representations of finite groups
  
  

The first is our reference for most of the course, and the second for the final chapter. The third is a useful reference.