0.1 Outline

The aim of this course is to study the representation theory of groups. If $G$ is a group, then a (finite-dimensional, complex) representation of $G$ is a homomorphism

for some $n\ge 0$. We can view this as an action of $G$ on ${ℂ}^n$. Two representations are equivalent if they are related by a change of basis. In the situations we will consider, every representation can be broken down into building blocks, called irreducible representations, and for a group $G$, there are two basic (related) problems:

1. 1.

   classify the irreducible representations;

2. 2.

   construct representions and decompose them into irreducibles.

The first half of the course will be devoted to the representation theory of finite groups. Here we will develop character theory: the character of a representation $\rho$ is the function $g\mapsto \mathrm{tr}\rho (g)$. It turns out that this function knows a lot about the representation — in some sense, it knows everything. We will develop methods to determine the irreducible characters of a finite group $G$. They fit into a “character table”, which has many beautiful properties.

The second half of the course focuses on the representation theory of certain groups of matrices — called linear Lie groups — such as ${\mathrm{SL}}_n(ℝ)$. In this case, we require that the representations $\rho$ are continuous. These groups are naturally smooth manifolds and their tangent spaces at the identity have the extra structure of a Lie algebra. We will explain how to relate Lie groups to their Lie algebras via the exponential map; we may then study the representation theory of Lie groups via the representation theory of Lie algebras. In the case of ${\mathrm{SL}}_2(ℂ)$ we will obtain an essentially complete understanding. In the case of ${\mathrm{SL}}_3(ℂ)$, we will make substantial progress. In both cases, the method is essentially to consider the action of the diagonal matrices, leading to the theory of weights.