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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.

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It's probably safe to say that beyond $n=2$ (and possibly $n=3$), little is known about the representations of these groups in characteristic 2. While there is a lot of general theory aimed at organizing such things for all reductive groups and at least most primes, the small primes pose extra problems at every step. On the other hand, the general theory (as in Jantzen's foundational book Representations of Algebraic Groups) tends to start over an algebraically closed field, then eventually specialize to smaller fields including finite ones. For small $n$ and $p$ there are some scattered computations in the literature but little in the way of a pattern. Indeed, for a prime like 2 this overlaps with the difficult problem of describing modular representations of symmetric groups.

While your results may be effective (using something like brute force), it's not easy to reconcile concrete methods including computer use with the urge to find theoretical patterns. Still, it's good to investigate the questions by whatever means are possible, in the absence of general answers.

ADDED: I tried to summarize what was known at the time (at least in the algebraic group literature) in Chapter 19 of my LMS Lecture Note Series 326 Modular Representations of Finite Groups of Lie Type (2006). I was aware of some work by algebraic topologists as well as the long tradition involving Dickson invariants and some more recent ideas such as Steenrod operations. But I may have overlooked some relevant papers. Steinberg's Lie-theoretic classification provides at least an outline of what is possible over finite fields, in terms of highest weights. But the details are tricky to fill in.

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