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A little background: I am currently learning a bit about category theory. I studied basic representation theory last semester (definitions, Maschke's theorem and Schur lemma, character tables, ... for finite groups and vector space over $\mathbb{C}$). In order to understand the material I am reading from Tom Leinster's book (Basic Category Theory), I decided to reformulate some definitions from my lectures about representation in the language of category theory (examples below).

For example, in my notes about representation theory, we define a linear representation as a homomorphism from a group $G$ to $GL(V)$ for some vector space $V$. As a basic first example, we can see a linear representation as a functor from $G$ (seen as a category with only one element) to the category of vector spaces over a field $k$.

Then, we often define the notion of isomorphic linear representation. In my lecture notes, it was defined as a $G$-equivariant isomorphism. If I am not mistaken, we can say the two representations $F : G \rightarrow \mathbf{Vect_k}$ and $H : G \rightarrow \mathbf{Vect_k}$ are isomorphic if $F \cong G$ in the category $[G, \mathbf{Vect_k}]$ (ie, there is some invertible natural transformation between $F$ and $H$).

It is my very first time reading about category theory, so I might say some absurdities. In my current understanding of the subject, category theory is mainly about finding patterns in seemingly different objects, and therefore, does not permit to understand every little detail (because it is there to generalize concepts).

So my question is: how far can I go into rewriting representation theory in terms of category theory objects? Maybe (big question mark), Schur lemma or Maschke's theorem might be possible to rewrite, but for more advanced results, I wonder...

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    $\begingroup$ I don't know much in representation theory, but I remember Frobenius reciprocity is a consequence of adjunction. $\endgroup$ – user171326 Jan 16 '17 at 16:36
  • $\begingroup$ Nice question. It's curious that I've studied category theory before representation theory, which I'm gonna study this year, and so I hope someone answer your question. $\endgroup$ – Xam Jan 16 '17 at 16:44
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    $\begingroup$ As an aside, one big thing you can do from this pov is that it suggests a notion of linear representation for categories, since $[\mathcal{C}, \mathbf{Vect_k}]$ makes sense for any small category. I believe representations of quivers can be seen as representations of categories. $\endgroup$ – Hurkyl Jan 16 '17 at 19:21
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RE: the comments: representations of groups and quivers are both special cases of representations of categories. If $R$ is a ring, then an $R$-rep of a group is a rep of its $R$-group algebra; an $R$-rep of a quiver is a rep of its $R$-path algebra; you can view an $R$-algebra as an $R$-linear category with one object; and the category of modules over an $R$-algebra is an $R$-linear (typically abelian) category.

Everything you see in a first course in representation theory can be encoded in category theory, if you really want to. Maschke's theorem says that (if $R$ is a field of characteristic coprime to the order of $G$) the category $C$ of $R$-reps of $G$ is semisimple. There's a pairing on the objects of $C$ defined by $\langle W,V\rangle= dim\ Hom_C(W,V)$, and this is the usual pairing on characters. Frobenius reciprocity is the fact that restriction and induction are adjoint, as mentioned in the comments. The other main results have similar interpretations.

In such a basic setting, this is all very uninteresting, but much of current research in representation theory involves understanding properties of categories of representations. The reason that you never really see people go to the effort of doing semisimple reps of finite groups categorically is that there isn't much reason to. The strength of category theory is really in organising abstract results, and suggesting ways that you might study some new objects that you might not know very much about. I'd suggest that the better way to think about things might be to treat translating the main theorems of basic representation theory of groups into categorical language, and then having this category as an excellent example of an abelian category. This can then give you a good toy example for the general theory!

Edit: I should say that the abstract rep theory of finite groups over "good" fields is "completely categorical", but this is only when you look at "general facts", by which I mean general theorems true for any finite group. But most research focuses on specific (classes of) groups, e.g. finite groups of Lie type. There are lots of phenomena which might not have such a clean categorical interpretation (e.g. reducibility of parabolic induction, Deligne--Lusztig theory); typically these kinds of results imply strictly weaker categorical results (in these examples, they tell you about block decompositions of categories), because they include non-categorical information -- they give explicit representatives of isomorphism classes of objects. In other words, once you want to get your hands dirty and do some useful representation theory, you'll probably have to leave the world of category theory!

Also, a couple of words of warning: once you leave the world of finite groups, or once you allow your coefficient ring $R$ to become sufficiently "bad", things can get more difficult; you start to encounter topological issues, which might make the category theory a bit more unpleasant.

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