Review of Group Representation Course

Main references are Lang and Serre.

Week 1

Differences between Ring, Matrix, and Endomorphism

Why do we care about Ring?

Definition

Group Ring, Simple Module, Idempotent

Useful Lemma

Weddenburn, Schur’s lemma

Block Decomposition

Week 2

Semi-simple Module and Ring

We often working on Semi Simple module or ring. Because it has a rich structure and theorem like Artin-Weddenburn Theorem.

To determine the ring is Ring, we can use the Group Ring.

Break in Simple

Main goal is break the representation space in simple modules. So, I think find the projection operator is quite natural.

To find the projection operator, we learn the following concepts. Commutant(Algebra)

Density Theorem

Representing action as a matrix(?)

Burnside’s Theorem

Week 3

Through the Application, we can use the character theory for the representation with field $k$ such that $char(k)=0$.

We learnt Ring earlier. But what about Simple Ring?

We can characterize simple ring using Artin-Weddenburn.

Also, through the Proof, we can find Non Trivial Central Idempotent through finding nontrivial central idempotent.

We have a remark, for internal case, External and Internal Decomposition is not a direct sum of rings. It is a coproduct of rings.

Week 4

I asked to my professor.

“This is a Group representation course. But why do we keep studying Ring and Module?” His answers were Why do we care about Ring? note and sometimes it is more easy to handle with module with a group ring. It has fluent tools to handle it.

The following table is saying about correspondences between group representation, $kG$-module, and characters.

We have one remark that Group Representation and $kG$-modules are equivalent. However, about $kG$-modules and characters, $kG$-modules can always be written in character theory, but converse holds only for $char(k)=0$.

Correspondence	Group Representation $\rho :G\to G{L}_k(V)$	$kG$-module	Character
Irreducibility	Irreducible	Simple Module
Isomorphism	Intertwine Operator	Module Isomorphism	Same characters
Sub object	Subrepresentation	Submodule
Decomposition	Block Decomposition	Direct Sum	Addition in $k$
	Tensor Product	Tensor Product	Product in $k$
	Subrepresentations	Submodules
	Group Representation	Dual Space	Dual Character
		Hom