I’d just like to expand a little on the topic of the irreducible characters being a basis for the class functions of a group from two times ago.
Let’s put an inner product on . Suppose . Then define .
The proof of the day is that the irreducible characters actually form an orthonormal basis of with respect to this inner product.
Let . Then we have that (although just a straightforward calculation, it is not all that short, so we’ll skip it for now). Thus .
So now examine
Where we note that since is a character . Thus we have that .
This fact can be used to get some neat results about the character table of a group, and as consequences of those we get new ways to prove lots of familiar things, like where the are the degrees of the characters. You also get a new proof of Burnside’s Lemma. I’m not very interested in any of these things, though.
I may move on to induced representations and induced characters. I may think of something entirely new to start in on. I haven’t decided yet.