Proving results in representation theory from category theory

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.

Solution 1:

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 $H \leq G$).