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Hello,

How can I solve the following question:

Let $G$ be a finite group and $\rho$ any non-trivial representation of $G$ over an arbitrary field. Show that $\sum_{g\in G}\rho(g)=0$.

Thank you!

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 This is false. MO is also not for questions of this type; have you read the FAQ? – Qiaochu Yuan Jul 10 2010 at 0:19
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 You need to assume $\rho$ is irreducible. This looks like a homework problem to me. – Agol Jul 10 2010 at 0:21
I'm sorry, the correct statement is "non-trivial irreducible representation". Is this question too trivial for MO? I think this is trivial when the field is C, but I have no idea how to solve this question for an arbirary field... – steve Jul 10 2010 at 0:26
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 You don't need Schur's lemma or the orthogonality relations. (Since I think this is a homework problem, I'll leave it at that.) – Qiaochu Yuan Jul 10 2010 at 0:56
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 For better or worse I stared at this for several minutes, so I will give a **hint**: you have a linear endomorphism $M$ of a vector space $V$ that you want to show is zero. For this is is sufficient to show that $Mv = 0$ for all vectors $v \in V$. Observe that $Mv$ lies in the subspace $V^G$ of $G$-invariant elements, and compute this subspace under your hypotheses. – Pete L. Clark Jul 10 2010 at 4:12
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closed as off topic by Qiaochu Yuan, José Figueroa-O'Farrill, Agol, Ryan Budney, Andy Putman Jul 10 2010 at 0:25

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