Point taken. This post wasn’t intended to be a beginner’s introduction; I very much had in mind an audience who had seen the basic results at least once, and I wasn’t trying to motivate interest in the subject. Hopefully my own motivation regarding the representation theory of the symmetric groups will become clear in a few posts.

]]>I think you have to consider the bifunctor given by (and similarly for ).

]]>I have a vector space V and I look at the functor which is tensor by V, the representation I get on will be trivial on the V and whatever it was on W. How do you get an arbitrary action on V to carry over via a tensor product functor?

]]>Thanks for the comment! That’s the part I was wondering about – the left/right convention was taken from the Wikipedia article about tensor products of modules, but it seemed out of place here after I thought about it. I’ll correct that part.

]]>I’m bothered by the second paragraph under “A proof for general representations”. The problem is the left/right thing. The category of _left_ G-reps is monoidal, where on objects the monoidal structure is \otimes_\CC and the action of g on V\otimes W is \rho_V(g) \otimes \rho_W(g), where \rho_V,\rho_W are the actions on V,W. Similarly, the category of _right_ G-reps is monoidal. In either case, the one-dimensional rep \CC with the trivial action is the monoidal unit.

You propose an a tensor product that multiplies a right G-rep with a left one. This will not define a representation of G (unless G is abelian, whence “right” and “left” are the same). What you can do is this: if R is any ring (perhaps \CC[G], the group algebra), and V is a right R-module and W is a left R-module, then the product V \otimes_R W is defined. However, it is not a module over R at all — the R action has canceled out. If you want an R-linear monoidal category, the easiest thing to do is to take the category of R-R bimodules, i.e. spaces with both a left R action and a right R action. (If R is commutative, these can be required to be the same, or not, depending on your goals.) The morphisms are required to commute with both actions, and the two actions are requires to commute. This category is monoidal with \otimes_R as the monoidal structure, and R with its canonical left- and right- actions on itself is the monoidal unit.

Ok, so it looks like you wanted that right representation in order to say that \Hom(V,W) = V^* \otimes W. The trick is this. If V is a left module over R, then V^* is a right R-module with the transpose action. How do you turn a right R-module into a left R-module? In general you cannot — doing so functorially requires R to have an antiautomorphism, i.e. an additive map s: R\to R satisfying s(ab) = s(b) s(a).

Fortunately, in the category of representations of a group, you have one of these: the map G \to G sending g \mapsto g^{-1} extends to an antiautomorphism of the group algebra \CC[G]. So this lets you define on V^* a _left_ G-action, namely g \mapsto \rho(g^{-1})^*, where \rho(g) is the action of g on V.

If you’re doing Hopf algebras, the map s is called the “antipode”. There is one more condition on the antipode in a Hopf algebra: it is required to be an antiautomorphism of the comultiplication as well as of the multiplication. This assures that (V\otimes W)^* = W^* \otimes V^*, which it should be. In the group algebra, the comultiplication is almost trivial, and so this condition is free.

Anyhoo, so all of this defines on \Hom_\CC(V,W) the structure of a left G-module, if V and W are both left G-modules. And then the action is that g acts on a linear map \phi by taking it to \rho(g) \circ \phi \circ \rho(g^{-1}), so you’re exactly correct that the G-intertwiners \Hom_G(V,W) are precisely the fixed points of the G-action \Hom_\CC(V,W).

A good interpretation of this is as follows. \CC with the trivial action is the monoidal unit in G-rep, and \Hom_G(\CC,V) is the vector space (not naturally a G-rep) of fixed points of the G-action on V. If we replace the word “G-rep” by, say, “sheaf”, then \Hom_G(\CC,V) is “the space of global sections of V”; this is a useful notion in any closed monoidal category, and the discussion above shows that G-rep is closed. So we’re saying that the global sections of the enriched \Hom(V,W) comprise precisely the set of morphisms \Hom(V,W). This is true in an arbitrary closed monoidal category.

]]>Ditto for Serre, which means I forget the proof quickly. Fulton and Harris reduce to the case where one of the irreducible modules is 1, which I think is an improvement.

I think Lang’s Algebra proves them a bit more functorially though.

]]>I wouldn’t trust my definitions to be exactly right, since I’m not working off of a reference here; in particular I may be somewhat confused about the tensor product.

]]>I agree. For example, Artin’s proof of the orthogonality relations involves a lot of juggling around terms in sums, which I’ve never found a satisfying way to convince myself of anything. To my mind, the categorical perspective is just a generalization of the idea that binomial coefficient identities, for example, should be proven bijectively.

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