Let $g$ be any non-identity element of a group $G$. How do you show that $G$ has an irreducible complex representation whose character $χ$ satisfies that the real part of $χ(g)$ is negative?
This question is from the book A Course in Finite Group Representation Theory by Peter Webb, Chap 3 Exercises 3. Any hint is appreciated.
That's what I have tried: let $χ_1,χ_2,...,χ_r$ be all irreducible complex representations of G. If we can prove that there is some $χ_i(g)$ having non-zero real part, then by Corollary 3.3.7. in that book the statement is clear.
Let $g\in G$, with $g\ne e$. By orthogonality of the character table $$0=\sum_\chi\chi(g)\overline{\chi(e)}$$ where the summation is over the irreducible characters. Now $\chi(e)=d_\chi$, the dimension of $\chi$ so that $$0=\sum_\chi d_\chi\chi(g).\tag{1}$$ The term corresponding to the trivial character in (1) is positive. Therefore there must be a term in (1) with negative real part.