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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?2011-05-03
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    Should "smallest irreducible" be "smallest faithful"?2011-05-03
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    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.2011-05-03
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    Complex representations are homomorphisms from $G$ to ${\rm GL}(n,{\bf C})$, not ${\bf F}_p$, right?2011-05-04
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    @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,2011-05-04
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    @Matt, OK, thanks. I note that the only information about $G$ that's needed in your answer below is that it's finite.2011-05-04

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