# Irreducible representation with trace zero in positive characteristic

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.

• 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 $F_2$. Dec 12 '16 at 10:44
• @Kyle Right, no $p$-group can provide an example since all their irreducibles are $1$-dimensional in chap $p$. Dec 12 '16 at 10:54
• This question has been previously asked and answered on MathOverflow: mathoverflow.net/questions/252855/… Dec 12 '16 at 13:23