I am re-posting a question I asked on math.se here because I am unsatisfied with the answers I obtained.

The irreducible modules of $\operatorname{GL}_n(\mathbb C)$ over $\mathbb C$ are completely classified and well-understood via Schur-Weyl duality, the algebraic Peter-Weyl theorem and the entire theory of reductive groups in characteristic zero.

I am looking for a good reference on what is known about the representation theory of $\operatorname{GL}_n(\mathbb F_p)$ over $\mathbb F_p$, i.e. the study of $\mathbb F_p$-vector spaces with an action of $\operatorname{GL}_n(\mathbb F_p)$. Here, $\mathbb F_p$ is the field with $p$ elements, $p$ a prime number. Are the irreducible modules completely classified? Can they somehow be indexed by certain partitions, similar to the characteristic zero case?

I am particularly interested in whether or not there is some equivalent of the Pieri rule, i.e. decomposing the tensor product of an irreducible representation with a symmetric power of $\mathbb F_p^n$. However, I suppose that question only makes sense when it is possible to classify irreducibles in some combinatorial way.