Thread: Image of a group's center under an irreducible representation

For an irreducible representation $\rho$ of a finite group $G$ what can be said about the image of the center $Z$ of $G$?

I have run into several problems telling me to apply Schur's lemma to this but I have not had success with this approach. Any help is appreciated.

Originally Posted by paulk

For an irreducible representation $\rho$ of a finite group $G$ what can be said about the image of the center $Z$ of $G$?

I have run into several problems telling me to apply Schur's lemma to this but I have not had success with this approach. Any help is appreciated.

you didn't menstion but i guess your base field is well, this is what we can say: where is the identity map. the reason is this:

let and then for all and we'll have that means is a module homomorphism.

also, since is algebraically closed, has an eigenvalue, say thus is a module homomorphism which is not an isomorphism because it has a non-zero kernel.

thus, since is irreducible, we must have by Schur's lemma.

Yes its over the complex numbers. Thanks for your help.