The following question arose in a research project, and I'm sure it must be well known. I even know a very indirect proof, and would love to know if anybody knows a simple one.
Here is the question. Let $G$ be a finite group and let $\rho$ be an irreducible representation of $G$, with dimension $d_\rho$. Let $\chi$ be the character of the representation. Then the problem is to show that $$ |\chi(g)| \le d_\rho $$ for all element $g \in G$.
Any help or reference greatly appreciated! Thanks, Nathanael