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My question here is primarily a reference request. I know a fair amount of representation theory of finite groups. I came across this question

How far can I develop representation theory from category theory?

about a connection between representation theory and category theory. In particular the nice answer given makes the point (if I am reading it correctly) that some results in representation theory are consequences of results in category theory. For example Frobenius reciprocity follows from "adjunction". I have been searching for a reference to this. I am comfortable talking about categories and functors, but I know only what I need to survive.

I would like to see a reference giving some details of how one might prove things like Frobenius reciprocity from category theory. I would, of course, also be happy with an answer that gives some details.

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  • $\begingroup$ Is the point of the bounty to attract an answer giving a category-theoretic proof of Frobenius reciprocity? $\endgroup$ – Stephen Feb 13 '18 at 21:28
  • $\begingroup$ @stephen There had been some comments early one saying that maybe the current answer wasn't correct. So I just wanted to attract more attention. If you think you could outline a proof of sorts of Frobenius using Category theory (or some other result in Representation theory, then I will give you the bounty. $\endgroup$ – Thomas Feb 14 '18 at 23:55
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    $\begingroup$ OK, well, I did my best to indicate how the character-theoretic version of Frobenius reciprocity may be obtained from the more general ring-theoretic version. Also, since the comments have since been deleted, it’s possible you missed the conclusion of that conversation (which happened in the comments to my answer): the commenter agreed that it was likely that Frobenius reciprocity can not be obtained in an efficient way from category theory ... cont’d ... $\endgroup$ – Stephen Feb 15 '18 at 14:44
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    $\begingroup$ ... I am in general reluctant to make assertions of the kind “There is no way to derive X using theory Y,” but in this case I believe I am on pretty firm ground. Of course, it would be possible to, say, prove using category theory that a unit-counit adjunction always produces an adjunction in the sense of my answer, and to then produce a unit-counit adjunction for Frobenius reciprocity. $\endgroup$ – Stephen Feb 15 '18 at 14:47
  • $\begingroup$ @Stephen: Thank you! When I wrote my comment above I hadn't realized that you were the one who had written the answer. I appreciate the time you took to write your answer and your edit. It was very helpful for me. $\endgroup$ – Thomas Feb 15 '18 at 16:39
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I would argue that it’s not quite right to say that one can prove Frobenius reciprocity using category theory. The statement of Frobenius reciprocity is an example of an adjunction: for a homomorphism of rings $R \to S,$ an $R$-module $M$ and an $S$-module $N$ the map

$$\mathrm{Hom}_R(M, \mathrm{Res}(N)) \to \mathrm{Hom}_S(S \otimes_R M, N)$$ given by $\phi \mapsto \psi$ with $\psi(s \otimes m)=s \phi(m)$ is an isomorphism of bifunctors. This is an adjunction $(S \otimes_R \cdot, \mathrm{Res})$. But one proves this using basic ring theory (it’s not hard: write down the inverse map). Applying this to groups rings and taking characters gives the character-theoretic version (see below for details).

A more serious example of using category theory to prove a statement in representation theory is Chuang-Rouquier’s proof of Broue’s abelian defect group conjecture for symmetric groups. This has become the prototype for applications of category theory in representation theory. You cannot go wrong by reading the original paper:

http://www.math.ucla.edu/~rouquier/papers/dersn.pdf

The idea is a typical one in modern mathematics: to prove something, first generalize it is a way that simultaneously rigidifies the problem while also placing it in a larger universe where more tools can be brought to bear.

Here are the details promised above: first, the inverse map is $$\psi \mapsto \left(m \mapsto \psi(1 \otimes m)\right).$$ Secondly, to obtain the character-theoretic version of Frobenius reciprocity we suppose we have a finite group $G$ and a subgroup $H \leq G$. Take $R=\mathbf{C} H$ with its natural inclusion in $S=\mathbf{C} G$. Now use the following facts:

(1) For a $\mathbf{C}G$-module $X$, the character of the dual space $X^*$ is the complex conjugate of the character of $X$, and if $Y$ is another $\mathbf{C}G$-module, the character of $X \otimes_\mathbf{C} Y$ is the product of the characters of $X$ and $Y$.

(2) With notation as above, the canonical isomorphism $$X^* \otimes_{\mathbf{C}} Y \to \mathrm{Hom}_{\mathbf{C}}(X,Y)$$ of vector spaces restricts to an isomorphism $$(X^* \otimes_{\mathbf{C}} Y)^G \to \mathrm{Hom}_{\mathbf{C}G}(X,Y).$$

(3) For a finite group $G$ and a $\mathbf{C} G$-module $X$, the operator $e=\frac{1}{|G|} \sum_{g \in G} g$ is the projection on the sub-space $X^G$ of $G$-fixed points, and its trace is therefore the dimension of $X^G$.

(4) Combining (1), (2), and (3), and writing $\chi$ and $\psi$ for the characters of $X$ and $Y$, we see that $$\mathrm{dim}_{\mathbf{C}}\left(\mathrm{Hom}_{\mathbf{C}G}(X,Y) \right)=\frac{1}{|G|} \sum_{g \in G} \overline{\chi(g)} \psi(g).$$ (This is, in my opinion, the right way to motivate the definition of the inner product of two class functions on $G$.)

(5) Apply (4) to the ring-theoretic version of Frobenius reciprocity given above to obtain the character-theoretic version (optional: use the definition of induction to give a formula for the character of the induced module in terms of the original character and the structure of the embedding $H \leq G$).

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  • $\begingroup$ I have been meaning to read that for a while, as it keep being highlighted as one of the original motivations for "classical" (i.e. not using 2-categories) categorification. $\endgroup$ – Tobias Kildetoft Feb 6 '18 at 17:10
  • $\begingroup$ A couple of other nice examples of use of category theory (i.e. categorification) in representation theory: Elias-Williamson's proof of Soergel's conjecture (and the first algebraic proof of the KL conjecture) arxiv.org/abs/1212.0791. Achar-Makisumi-Riche-Williamson's proof of character formulas for tilting modules arxiv.org/abs/1706.00183. And just because I am the one writing: My classification with Mazorchuk of projective functors on parabolic category $\mathcal{O}$ in type $A$ arxiv.org/abs/1506.07008 $\endgroup$ – Tobias Kildetoft Feb 7 '18 at 7:03
  • $\begingroup$ @TobiasKildetoft The paper of Chuang and Rouquier does use $2$-categories (without formalizing it) and in fact most of the main ideas of Rouquier's paper $2$-Kac-Moody algebras are already present there. Indeed, the difference between what they call a "weak" $\mathfrak{sl}_2$-categorification and a $\mathfrak{sl}_2$-categorification is precisely the difference between categorification (replacing vector spaces with abelian categories and linear maps with functors) and $2$-categorification (in addition, keeping track of certain natural transformations). $\endgroup$ – Stephen Oct 27 '18 at 20:33

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