Decomposing a reducible representation

[Gulli]

Homework Statement

In order to construct a character table (or to solve problems that directly ask for irreducible representations) I have to be able to decompose a representation into irreducible representations. However I don't know how to do this in general.

Homework Equations

I understand the definition of an irreducible representation and know it amounts to converting the atrices of the representation into blocks.

The Attempt at a Solution

I have one example available (D3) which is solved by finding transforming the matrices using a transformation matrix consisting of an invariant vector (for example (1,1,1)) and it's orthogonal complement. I doubt this works in general though, or if it does it can get very tedious, but I may be wrong. So, is this the general method, if not, what is (an example to demonstrate would help)?

 

[Klockan3]

First you find the conjugacy classes, then you write up the character table, then you find the character for your representation which gives you what the representation should be reduced to since the characters for the classes forms an orthonormal basis.

 

[Gulli]

First you find the conjugacy classes, then you write up the character table, then you find the character for your representation which gives you what the representation should be reduced to since the characters for the classes forms an orthonormal basis.

I know characters can tell you how many irreducible representations there are and what dimensions they have (provided you know the amount of conjugacy classes, which sometimes takes a lot of work, but at least it's easy), but then you still have to find them and that's kinda where I'm stuck.

 

[Klockan3]

If you want to find the actual conjugation which represents the decomposition then you have no choice but to find the invariant subspaces. I'd say that the easiest way to do that would be to find the eigenvectors for the matrices, takes a bit of work but not too much and it should make it clear which subspaces are invariant.

 

[Gulli]

If you want to find the actual conjugation which represents the decomposition then you have no choice but to find the invariant subspaces. I'd say that the easiest way to do that would be to find the eigenvectors for the matrices, takes a bit of work but not too much and it should make it clear which subspaces are invariant.

Some excercises ask me to explicitly write down the irreducible representations (convert the matrices to block form). Others ask me to write down the character table (consisting of the character of the standard representation and the characters of the irreducible representations). In both cases I need to know what the irreducible representations look like (write down their matrices). I have one example for D3 where an invariant vector (invariant under all possible operations of the group) and it's orthogonal complement are used to construct a transformation matrix. I don't understand what the eigenvectors have to do with it (although an invariant vector is an eigenvector with eigenvalue 1 (for the matrix of every element of the group), but what if there is no such eigenvalue, or would that be a sign the representation is irreducible already?)

 

[Gulli]

P.S. I can see how eigenvectors would be useful in the special case of abelian groups because they have only 1-dimensional irreducible representations so finding those would be a simple matter of diagonalizing the matrix using the the eigenvectors.