Group character

Finite group character values

Let be a finite group, be a group representation, affording character . Then for all , where are the cyclotomic integers and . rep Moreover,

1. ;
2. iff is a homothety;
3. iff is the identity.

Proof

The eigenvectors of must have eigenvalues of finite order dividing , and thus are each powers of . It follows that , which is equal to the sum of these eigenvalues, is in .

It follows that is precisely the kernel of any representation affording .

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