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