# Part II - Representation Theory

## Lectured by S. Martin, Lent 2016

These notes are not endorsed by the lecturers, and I have modified them (often significantly) after lectures. They are nowhere near accurate representations of what was actually lectured, and in particular, all errors are almost surely mine.

This is an HTML version of the notes, generated using some horribly-written scripts and pdf2htmlEX. These are not guaranteed to display well, but do let me know if something is broken. Note however that I cannot help you if your browser does not support standard HTML features (eg. this part is known not to work well with w3m). You can either view all sections in a single page (Full version), or access individual sections below. If you want to download a pdf, head to the Notes page.

# Contents

- V Full version
- 0 Introduction
- 1 Group actions
- 2 Basic definitions
- 3 Complete reducibility and Maschke's theorem
- 4 Schur's lemma
- 5 Character theory
- 6 Proof of orthogonality
- 7 Permutation representations
- 8 Normal subgroups and lifting
- 9 Dual spaces and tensor products of representations
- 9.1 Dual spaces
- 9.2 Tensor products
- 9.3 Powers of characters
- 9.4 Characters of G x H
- 9.5 Symmetric and exterior powers
- 9.6 Tensor algebra
- 9.7 Character ring
- 10 Induction and restriction
- 11 Frobenius groups
- 12 Mackey theory
- 13 Integrality in the group algebra
- 14 Burnside's theorem
- 15 Representations of compact groups