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I've been wanted to learn some basic theories of the (non-semisimple) representation of the finite group over a finite field.

I have been guessing that the materials might be contained in the books on $\textbf{the modular representation theory}$.

So I once tried to read the recent book of Peter Schneider.

But it seems that in the book, lots of background materials are presented, the theories on the complete local commutative integral domain, theory of blocks and so on.

What I want to say is that maybe the modular representation theory is deep and difficult, so those prerequisite algebraic theories might be necessary. The problem is that I cannot be sure of whether I can find a tool useful in studying the representation of finite group over $\textbf{a finite field}$.

Now I can also guess that the 'concept' of representation theory over finite field is elementary but very difficult in 'practice', as usual in most field of mathematics.

I think the representation over a finite field is more natural and maybe more interesting (even though it might be naive) than the representation over complete local field. Thus I think there might be other mathematicians who are not specialists in modular representation theory and have the same question.

'Does the modular representation theory give us a tool to study the representation theory of finite group over a finite field?'

If so, then is there some direct reference that focuses on some (even though past) application to the representation over finite field not covering the hot and deep theoretical issues of the modular representation theory?

Thank you very much!

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    $\begingroup$ Honestly speaking, I cannot understand the slightest bit of what you're asking, but I have a feeling that this www.researchgate.net/publication/268247317_Two_More_Variations_on_the_Triangular_Theme paper may be somewhat interesting for your taste. $\endgroup$
    – Denis T
    Commented May 11, 2021 at 7:44
  • $\begingroup$ @DenisT. Thank you very much for your interest. I just wanted to know 'in advance' whether the theories of the modular representation theory is and can be used to study the representation over finite field? (Maybe by using the fact that for any finite field F, there is a local field whose residue field is F?) $\endgroup$
    – gualterio
    Commented May 11, 2021 at 7:54
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    $\begingroup$ There are some groups, for example, the symmetric group, where the representation theory is mostly the same over $\mathbb{F}_p$ as over its algebraic closure. But still it is sometimes necessary to use the algebraic closure or to work with representations defined over rings such as the $p$-adic integers $\mathbb{Z}_p$ in order to use more general results. For example idempotent lifting and the $cde$-triangle are useful in the symmetric group and need the theory of $p$-modular systems. $\endgroup$
    – Mark Wildon
    Commented May 11, 2021 at 10:36
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    $\begingroup$ To be honest I'm not an expert on this. Benson has a two book series on representation theory and group cohomology. The first book does group representation theory and the second uses it in group cohomology. The first book uses all the fancy machinery and can be a bit brief if you have never seen this stuff. You might flip through the second book on cohomology to get an idea how the first book machinery is used to see why it is useful $\endgroup$
    – Benjamin Steinberg
    Commented May 11, 2021 at 13:16
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    $\begingroup$ I often use Benson's two books as a reference. But I don't think they are a good place to learn the material. There are three paths to modular representation theory: character theoretic (Brauer characters, blocks via orthogonality, e.g. Navarro's book), modules (vertices, Green correspondence, blocks, e.g. Alperin's book), rings (orders, number theory, idempotents, blocks, e.g. Puig's book or the two volumes of Curtis and Reiner, Methods of representation theory). Any might be to your taste, but Alperin's book Local representation theory is rightly admired and gets quickly to deep results. $\endgroup$
    – Mark Wildon
    Commented May 11, 2021 at 14:28

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