A character of a representation is the largest on the unity

Question

Let $G$ be a finite group and $\rho:G\rightarrow \textrm{GL}(V)$ be a representation of $G$.
Denote the character by $\chi_\rho$.

Is it correct to say that $\chi_\rho (e)\geq \chi_\rho (g)$ for all $g\in G$, where $e$ is the unity?
If so, how can we show it?

Answer 1

I assume $\rho$ is a representation on a $\Bbb{R}$ vector space for "largest" to make sense.

Letting $n=|G|$, $g^n=1$ so that each eigenvalue of $\rho(g)$ is a $n$-th root of unity. ie. $\det(X-\rho(g))=\prod_{j=1}^{\dim V} (X-a_j(g))$ and $Tr(\rho(g))=\sum_{j=1}^{\dim V} a_j(g)$ where each $|a_j(g)|=1$.

$Tr(\rho(g))$ is maximum when all the $a_j(g)=1$, in particular for $g=1$.