Is "real character" means self-conjugate for representation of finite groups

Question

For representation of finite groups on complex vector space, I knows two facts:

1. If a representation is self-conjugate (it is equivalent to its complex conjugate), then its character function is real-valued.

2. A representation is determined by its character function.

So I wonder whether a real-valued character function means that this representation is self-conjugate. But I cannot give a proof now.

Is this true? Or, can you give a counter-example?

Answer 1

I think I found a way to prove this.

First, if it is self-conjugate, clearly the character is real-valued. On the other hand, if the character is real-valued, then by \begin{equation} \chi_{D^*}(g) = \operatorname{Tr}(D^*(g)) = \operatorname{Tr}(D(g))^* = \chi^*_{D}(g) = \chi_{D}(g) \end{equation} The conjugate representation has the same character, hence is equivalent to original representation. Hence this representation is self-conjugate.