Let $G$ be a finite $p$-group. What are irreducible representations of $G$ over a field of characteristic $q$, such that $(p,q)=1$ ? Can we say something in general ? In particular, if there exists some technique to find those explicit representations in case of small groups of order $p^3, p^4, p^5 ...$ ?
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2$\begingroup$ The first question is standard representation theory. The irreducible representations in coprime characteristic over algebraically closed fields can be realized directly as reductions from those in characteristic zero. Over finite fields, things are a little easier, because the Schur index is always trivial. For the second question, there are algorithms for computing irreducible representations, but the question needs a more focus. $\endgroup$– Derek HoltJul 22, 2021 at 7:38
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For the last part of the question, they are all induced from primitive representations of subgroups, so you need to understand primitive representations of $p$-groups. If $G$ is a $p$-group with a faithful primitive (irreducible) representation over a finite field $F$ of coprime characteristic $q$, then all Abelian normal subgroups of $G$ are cyclic by Clifford theory. If $p$ is odd, this implies that $G$ itself is cyclic. If $p =2$, I think $G$ can also be dihedral, (generalized) quaternion or semidihedral, as well as cyclic.
If $p$ is odd, and $e$ is the least power of $q$ such that $p^{m}|(|F|^{e}-1)$, then a cyclic group of order $p^{m}$ has a faithful irreducible representation of degree $e$ (but no lower dimension) over $F$.
If $p =2$, the situation is more delicate: for example, a semidihedral group of order $16$ has a faithful $2$-dimensional primitive representation over $F$ when $|F| = 3$.
answered Jul 22, 2021 at 9:00