# Representation theory of the general linear group over a finite prime field

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?

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.

Edit: I am re-posting this at MO in the hope of getting some more details.

• It's important to distinguish here between the representations of the algebraic group $\mathrm{GL}_n / \mathbb{F}_p$, and representations of the finite abstract group $\mathrm{GL}_n(\mathbb{F}_p)$. Your notation suggests the latter, but your first and last paragraph suggest the former. – David Loeffler Oct 31 '14 at 14:08
• I worded my question more cautiously. I am not sure if I want the representations of the algebraic group $\operatorname{GL}_n/\mathbb F_p$, in case that somehow involves the algebraic closure of $\mathbb F_p$, I am indeed looking at the finite group $\operatorname{GL}_n(\mathbb F_p)$ acting on $\mathbb F_p$-vector spaces. – Jesko Hüttenhain Oct 31 '14 at 14:16
• @Thomas: Most definitely not, everything is over $\mathbb F_p$. – Jesko Hüttenhain Oct 31 '14 at 15:48
• It seems to me that the question is clear: the representations the OP is interested in are the representations of the finite group $GL_n(F_p)$ on finite dimensional vector spaces over the finite field $F_p$. So $\mathrm{det}^{p-1}$ would be trivial. Or am I being dense? – Stephen Oct 31 '14 at 17:52
• A representation of an algebraic group $G$ is a morphism of algebraic groups $G\to GL_n$. The representation $\det^{p-1}$ is not trivial in that sense (as it is not trivial on extensions of $F_p$) – Mariano Suárez-Álvarez Nov 4 '14 at 18:58