# Representation theory. Illustration of a theorem

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.

• Irreducible representation of what? There is more than one theorem, see here. Apr 26 '19 at 11:51
• I know this book and don't have to look at chapter 2.3. You should add representations of (finite) groups in the post. Apr 26 '19 at 12:20

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).