Finite group character values

Let $𝐺$ be a finite group, $Γ : 𝐺 \to ℂ 𝑑$ be a group representation, affording character $𝜓 : 𝐺 \to ℂ$ . Then for all $𝑔 \in 𝐺$ , $𝜓 (𝑔) \in ℤ [𝜁 𝑛]$ where $ℤ [𝜁 𝑛]$ are the cyclotomic integers and $𝑛 = | 𝐺 |$ . #m/thm/rep Moreover,

$| 𝜓 (𝑔) | \leq 𝜓 (1)$ ;
$| 𝜓 (𝑔) | = 𝜒 (1)$ iff $Γ (𝑔) \in Z (ℂ 𝑑)$ is a homothety;
$𝜒 (𝑔) = 𝜒 (1)$ iff $Γ (𝑔) = 1 𝑑$ is the identity.

Proof

The eigenvectors of $𝜓 (𝑔)$ must have eigenvalues of finite order dividing $𝑛$ , and thus are each powers of $𝜁 𝑛$ . It follows that $𝜒 (𝑔)$ , which is equal to the sum of these eigenvalues, is in $ℤ [𝜁 𝑛]$ .

It follows that $k e r 𝜒 = {𝑔 \in 𝐺 : 𝜒 (𝑔) = 𝜒 (1)}$ is precisely the kernel of any representation affording $𝜒$ .

#state/tidy | #lang/en | #SemBr