anyone up on group theory?

[oswaler]

I'm just learning it, so this is a very basic question.

I understand how to construct the multiplication and character tables for a group. From the characters I need to construct irreducible representations. I have read the notes from the class, the textbook and a couple other books and they all do the same thing.

They show how to calculate representations and then say now we need to find the irreducible representations. To do that you need to construct the character table. They then show how to construct the character table and then stop. I can't find any description of how to use the character table to construct the irreducible representations or even an example of what an irreducible representation looks like.

If we take a very simple example, D3 - the symmetry rotations for a triangle. I understand the multiplications of the operations, I see that there are 3 classes based on cyclic operations and I calculated the character table so I have the characters for the irreducible representations. But now how do I construct the irreducible representations?

Any help would be greatly appreciated.

Thanks - Eric

[jemidiah]

I've never heard of an irreducible representation with regards to a group... the only thing that comes close is irreducibles in rings, but that's not nearly the same thing.

It sounds like you should ask your professor what the question really wants--sorry I can't be of more help.

[oswaler]

hmmm, I think the question is pretty clear. I know irreducible representations are a pretty important element of group theory.

If anyone out there can help I would still appreciate it.

Thanks - Eric

[jemidiah]

I Wikipedia'd irreducible representations, and got to representation theory--which is (apparently) a branch of group theory, but I'm not familiar with it at all. It looks interesting too....

Anyway, the only bit I could see that might be helpful is a brief example of an irreducible representation. I'm pretty sure it'll be nothing new, but on the off-chance it is, I thought I'd throw it out there:

From http://en.wikipedia.org/wiki/Group_representation

In the example above, the representation given is decomposable into two 1-dimensional subrepresentations (given by span{(1,0) } and span{(0,1)}).