Irreducible representation decomposition

Question

Any faithful finite group representation can be written as a sum of irreducible representations $\rho = \oplus_{i} a_i \rho_i$ such that $Ker(\rho)=0=\bigcap_i Ker(\rho_i)$ - is this sufficient to give us the smallest (degree) faithful representation? If $\bigcap_i Ker(\rho_i)=0$, I think we can not deduce $\forall i, Ker(\rho_i)=0$: my friend said to look at the kernels of the irreducible characters to find the smallest such $\rho$, but how to we achieve this? I can't see how knowing the kernels of the individual irreducible characters would be sufficient to find the smallest faithful representation, but maybe I'm missing some condition which follows from orthogonality or something similar. Thanks for the help!

Answer 1

I think that the point of your friends comment is the following: as you wrote, if $\rho = \oplus_i \rho_i,$ then $ker(\rho) = \cap_i ker(\rho_i)$. So you need to choose $\rho_i$ so that $\cap_i \ker(\rho_i)$ is trivial. If you know the kernels of the various irreps. $\rho_i$ in some explicit form, presumably you can choose a minimal collection of them so that $\cap_i \ker(\rho_i)$ is trivial.

(At least if we are talking about representations of a finite group over a field of characteristic zero, this is possible, since general theory assures us that a group has a faithful semi-simple representation, e.g. its regular representation.)