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.

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