representation theory of finite groups

Setup

In these notes, I consider to be a finite group. Here we consider the base field of all the representations to be since it is algebraically closed, though any other algebraically closed field would work too, if you are careful about characteristics.

Finite abelian groups

Let be a finite abelian group. Then has exactly many simple modules.

Proof

Since is abelian, is commutative. Therefore, using the decomposition from the Artin-Wedderburn theorem, we have that , since each of the matrices have to be commutative. This also gives that since the number of conjugacy classes of is .

Number of 1-dimensional representations

Let have commutator subgroup . Then has exactly 1-dimensional representations.

Proof

is abelian, so any group homomorphism factors through . That is

So this gives

so there can only be of them.

Examples

todo - some in lecture 28

Characters and dimension

Let be a representation. The multiplicity of the trivial representation in is

Proof

Define

Then

1. for all .

Thus, is a projection on the trivial summand of the decomposition where the dimension is

Characters and class functions

Let be the vector space of class functions on .

We can define a hermitian inner product on by

We can see that this is an inner product since

This gives us a nice new notation where using the tensor-hom adjunction,

leads to

Simple characters

The simple characters

form an orthonormal basis of .

Proof

todo - Lecture 31

Using this in decomposition

Let be finite dimensional representation of a finite group. Then is a module over , and it can be decomposed into simple modules

Using the orthonormal basis from above, this means that