# Sum of values of an irreducible character is non-negative integer

I am trying to prove the following fact:

If $$G$$ is a finite group, $$\chi$$ is a complex irreducible character of $$G$$ and $$\{g_1,\cdots,g_r\}$$ is a complete set of representatives of conjugacy classes in $$G$$ then $$\chi(g_1)+\cdots + \chi(g_r)$$ is a non-negative integer.

It is easy to prove that the above quantity is integer. Sketch of proof: if $$|G|=m$$ then the components in the above expression lie in $$\mathbb{Q}(\zeta_m)$$ and automorphisms of this field correspond to bijections from $$G$$ to $$G$$ which permute conjugacy classes (maps $$G\rightarrow G$$, $$g\mapsto g^l$$ where $$(l,|G|)=1$$ and correspondingly there are Galois automorphisms); hence above quantity is invariant under all automorphisms of $$\mathbb{Q}(\zeta_m)$$ over $$\mathbb{Q}$$ and is an algebraic integer.

I didn't get why it is non-negative? Any hint?

Hint: let $$G$$ act on $$G$$ by conjugation and consider the permutation character $$\pi$$ of this action. Show that the Frobenius product $$[\chi,\pi]$$ is exactly the sum you want to calculate.