characters of irreducible representations, orthogonality relation

Question

I Want to prove that it's impossible that to irreducible representation Xpi(g)≥0 for all g
(pi irreducible representations) I am thinking about Second orthogonality relation, and to use the fact that characters of irreducible representations, they are orthonormal. And inner product of two different representations equal to zero. I cannot arrange my thoughts and do not know if I am in the right direction, I would be happy if someone can help, and I would be happy if there is someone who knows a better solution.

Answer 1

Hint: if $\chi$ is irreducible and not the principal character $1_G$, then $[\chi,1_G]=0$. So $[\chi,1_G]=\frac{1}{|G|}\sum_{g \in G}\chi(g)=0$. Hence $\sum_{g \in G}\chi(g)=0$. But if $\chi(g) \geq 0$ for all $g \in G$, then $0=\sum_{g \in G}\chi(g)=\chi(1)+\sum_{g \in G-\{1\}}\chi(g) \geq \chi(1) \geq 1$, a contradiction.