Let $F_2$ denote the finite field of two elements, and $GL(n,2)$ the general linear group of degree $n$ over $F_2$. I have a promising line of inquiry into identifying the structure of simple $F_2\ GL(n,2)$-modules, but I am a topologist and I am not well-acquainted with the status of modular representation theory research. I have two questions for which I did not find answers in classic texts or with a literature search.

Question 1: Are the structures of simple $F_2\ GL(n,2)$-modules known? If so, where are they described?

If the answer is "no," then I may have something to contribute. It would be helpful to have an answer to the following:

Question 2: Given an isomorphism class of simple $F_2\ GL(n,2)$-modules (or equivalently a $2$-regular tableaux with longest row length $n$) what are the corresponding idempotents in $F_2\ GL(n,2)$?

The reason for my interest is that for each simple module $M$, I have an accessible module $N\cong M \oplus M^\perp$, and I can prove that $M$ is not a composition factor of $M^\perp$. A complete orthogonal set of idempotents would provide a basis for $M$.

Let me clarify what I am looking for by giving an ideal answer to my second question for $n=2$:

In $F_2\ GL(n,2)$, we can write $1=a+b_1+b_2$, where $a$,$b_1$ and $b_2$ are orthogonal primitive indecomposable idempotents. The idempotent $a$ corresponds to the trivial module $a$ (tableaux columns {1,1}), and $b_1$ and $b_2$ are correspond to the standard representation (tableaux columns {2,1}).

$a=1+\left( \begin{array}{cc} 1&1\\ 1&0 \end{array} \right)+ \left( \begin{array}{cc} 0&1\\ 1&1 \end{array} \right)$

$b_1=1+ \left( \begin{array}{cc} 1&1\\ 0&1 \end{array} \right) + \left( \begin{array}{cc} 1&0\\ 1&1 \end{array} \right) + \left( \begin{array}{cc} 1&1\\ 1&0 \end{array} \right)$

$b_2=1+ \left( \begin{array}{cc} 1&1\\ 0&1 \end{array} \right) + \left( \begin{array}{cc} 1&0\\ 1&1 \end{array} \right) + \left( \begin{array}{cc} 0&1\\ 1&1 \end{array} \right)$

In practice I would expect a formula or procedure for generating the idempotents; the above were discovered using brute force.