Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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

share|improve this question

1 Answer 1

$\chi(g)$ is the sum of the eigenvalues of $\rho(g)$. These are roots of unity, so have absolute value one, and there are $d_\rho$ of them. Thus the triangle inequality gives the inequality you want.

share|improve this answer
    
Thanks very much ! I realized as much on my way home. Sorry to have asked a trivial question ! Nathanael –  Nathanael Berestycki Nov 28 '11 at 18:28

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.