# Irreducible Character Basis

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 $cf(G)$ from two times ago.

Let’s put an inner product on $cf(G)$. Suppose $\alpha, \beta \in cf(G)$. Then define $\displaystyle \langle \alpha, \beta \rangle =\frac{1}{|G|}\sum_{g\in G} \alpha(g)\overline{\beta(g)}$.

The proof of the day is that the irreducible characters actually form an orthonormal basis of $cf(G)$ with respect to this inner product.

Let $e_i=\sum_{g\in G} a_{ig}g$. Then we have that $a_{ig}=\frac{n_i\chi_i(g^{-1})}{|G|}$ (although just a straightforward calculation, it is not all that short, so we’ll skip it for now). Thus $e_j=\frac{1}{|G|}\sum n_j\chi_j(g^{-1})g$.

So now examine $\frac{\chi_i(e_j)}{n_j}=\frac{1}{|G|}\sum \chi_j(g^{-1})\chi_i(g)$
$=\frac{1}{|G|}\sum \chi_i(g)\overline{\chi_j(g)}$
$= \langle \chi_i, \chi_j \rangle$.

Where we note that since $\chi_j$ is a character $\chi_j(g^{-1})=\overline{\chi_j(g)}$. Thus we have that $\langle \chi_i, \chi_j \rangle = \delta_{ij}$.

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 $|G|=\sum n_i^2$ where the $n_i$ 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.