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This is mostly a reference request ... I think.

I am a bit familiar with representation theory of finite groups. Here I have seen a representation as a homomorphism $\rho: G \to GL(V)$ where $V$ is a complex vector space. But, I am guessing that this would also make sense of $V$ was a vector space over another field.

My basic question is if this is "a thing"?

If this is a thing, I would like to see a reference on this. Are there books/notes that do all the same things (invariant subspaces, irreducible, Frobenius Reciprocity, ...) but just work with a vector space over a finite field?

(If this is not a thing, then why not? What happens that makes this uninteresting?)

John Doe
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There is a thing called modular representation theory where one considers fields $F$ of positive characteristic $p$ and groups $G$ of order divisible by $p$. By the converse of Maschke's theorem, the group algebra $FG$ is not semisimple. This makes things more complicated. To make things a bit easier, most books assume that $F$ is algebraically closed (in particular, not finite). Even in this case, the irreducible representations of familiar groups like symmetric groups are not known in general (not even their degrees). You may take a lot at the books of Curtis-Reiner, Feit or Gow et al. ("Representation theory in arbitrary characteristic").

Brauer Suzuki
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  • I assume that the Curtis-Reiner book here is the same as the one mentioned in the comment above? Do you have a link for the other two? – John Doe Apr 16 '22 at 16:16
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    I haven't seen this comment at the time of writing. Curtis-Reiner also wrote an extended 2-volume book called "Methods of representation theory I, II". – Brauer Suzuki Apr 16 '22 at 16:18