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Is there an example of an irreducible representation $\rho:G\rightarrow GL_n(F)$ where $char(F)>0$ such that $trace(\rho(g))=0$ for all $g\in G$?

Of course we have to consider $F$ and $G$ (finite group) with $F[G]$ non-semi-simple.

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  • $\begingroup$ I almost thought that $D_4$ over $F_2$ would be an example, since the standard two-dimensional representation is indecomposible over $F_2$, but it does not remain irreducible over $F_2$. $\endgroup$ Dec 12, 2016 at 10:44
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    $\begingroup$ @Kyle Right, no $p$-group can provide an example since all their irreducibles are $1$-dimensional in chap $p$. $\endgroup$ Dec 12, 2016 at 10:54
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    $\begingroup$ This question has been previously asked and answered on MathOverflow: mathoverflow.net/questions/252855/… $\endgroup$ Dec 12, 2016 at 13:23

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Just so this has an answer (I'll make the post community wiki since it's not my answer):

This answer on MathOverflow says that I.M. Isaacs, Character Theory of Finite Groups, Corollary 9.22, Dover, p. 155, states that every irreducible representation of a finite group over any field has a non-zero character.

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