Representation theory. Illustration of a theorem

Question

I read a book and it said:

Theorem 2.3 The characters of irreducible representations are orthonormal.

Can someone provide a detailed example to this theorem?

Thank you.

Answer 1

The easiest example would most likely be the trivial character and the signum on $S_n$. They are both irreducible, so the fact that they are orthogonal means that $$\sum_{\pi \in S_n} sign(\pi) = 0,$$ which should be relatively easy to show (assuming $n \geq 2$ of course).

As this is a central result in representation theory, I would check the book for further examples or exercises. The result will surely come up at a few more places, e.g. when determining character tables.