I am searching for an example of a finite group $G$ and an algebraically closed field $K$ of characteristic $p$, s.t. $p \nmid |G| $ with an irreducible representation $\rho : G \to \operatorname{GL}(V)$ and $p | \operatorname{dim}(V) < \infty$.
The question did come up since for $\operatorname{char}(K)=0$ we have the relation $\operatorname{dim}(V) \mid |G|$, so an example like above does not exist.
But how does it look like if we have an assumption like above?
Best Answer
If $K$ is algebraically closed and $char(K)$ does not divide $|G|$ then the irreducible representations over $K$ have the same dimensions as over $\mathbb{C}$. In particular the dimensions must divide $|G|$ and therefore not be divisible by $char(K)$.