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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.

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

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