### IV.9 Representation Theory

*Ian Grojnowski*

### 1 Introduction

It is a fundamental theme in mathematics that many objects, both mathematical and physical, have symmetries. The goal of GROUP [I.3 §2.1] theory in general, and representation theory in particular, is to study these symmetries. The difference between representation theory and general group theory is that in representation theory one restricts one’s attention to symmetries of VECTOR SPACES [I.3 §2.3]. I will attempt here to explain why this is sensible and how it influences our study of groups, causing us to focus on groups with certain nice structures involving *conjugacy classes.*

### 2 Why Vector Spaces?

The aim of representation theory is to understand how the *internal* structure of a group ...