# Irreducible representation decomposition

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!

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Representation of what as what? Do you mean complex representations of finite groups? Representations of Lie algebras? – Jack Schmidt May 3 '11 at 13:57
Should "smallest irreducible" be "smallest faithful"? – Jack Schmidt May 3 '11 at 13:57
Yes, sorry, corrected now! I wasn't aware you could have representations of Lie algebras, I meant complex representations of finite groups: in particular, groups G $\subset$ GL(n,$\mathbb{F}_p$) is what I'm most interested in. – S. Granther May 3 '11 at 19:22
Complex representations are homomorphisms from $G$ to ${\rm GL}(n,{\bf C})$, not ${\bf F}_p$, right? – Gerry Myerson May 4 '11 at 1:48
@Gerry: Dear Gerry, I think that the homomorphism is to $\mathrm{GL}(n,\mathbb C)$, but the group $G$ starts out life as a subgroup of $\mathrm{GL}(n,\mathbb F_p)$. (So the OP is interested in complex representations of subgroups of general linear groups of finite fields.) Regards, – Matt E May 4 '11 at 2:06

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.