# Reference needed for representation theory of direct products of algebraic groups over a field (of arbitrary characteristic)

In my dissertation I proved a certain theorem(s) concerning the representation theory of a direct product G x H of algebraic groups over a field, given those of G and H. But I would wager 100:1 that this theorem is well known, and as such I am looking for a primary reference for it. I have combed (my personal) literature for it, e.g. Jantzen, Humphreys, etc., but cannot find it.

The theorem(s) can be found in section 2 of the following pdf file:

http://homepages.utoledo.edu/mcrumle/references%20needed.pdf

Please ignore the other sections of the file as they pertain to questions having little relevance here.

Thanks in advance for any help.

$\underline{\text{Question, version 2.0}}$

Sorry again for the confusion, my notation appears to have been sloppy. And sorry that I haven't posted the entire thing on this page, but I really think it's more efficient (for everyone) that I post it here:

http://homepages.utoledo.edu/mcrumle/Reference%20Needed%20-%20Direct%20Products.pdf

This theorem I believe is much more basic than the gracious people who bothered to try to answer it (in my garbled terminology) have made it out to be. I am talking about arbitrary representations (not irreducible, etc.) over arbitrary fields (not algebraically closed, etc.) for arbitrary algebraic groups (not reductive, etc.). It is simply the statement that any $G \times H$-module structure on a $k$-vector space $V$ can be factored (as a product of matrices, or linear maps if you like) uniquely into a product of commuting representations for $G$ and $H$ on that same vector space. In section 2 there are no tensor products or direct sums of vector spaces taking place; the vector space $V$ is fixed throughout, and all three of the groups $G$, $H$, and $G \times H$ are acting on it.

The corollaries that follow in section 3 merely re-affirm the suspicion that, for fixed $G \times H$-representations on the fixed vector spaces $V$ and $W$, things like "tensor product", "direct sum", and "morphism" of $G \times H$-modules should behave as expected with respect to their respective $G$ and $H$ modules.

Thank you once again for any help.

• This question would be much improved if it were more self-contained. Presumably you have the TeX source of your own dissertation, so you can just copy and paste the statements of the theorems. – S. Carnahan Nov 10 '11 at 9:19
• There is something wrong with the way you formulate Theorem 2.1. If $(a_{ij})$ is a matrix acting on $V$, then $a_{ij}\otimes 1$ is, for me, a matrix acting on the tensor product of $V$ and something else, not on $V$. – Vladimir Dotsenko Nov 10 '11 at 9:50

I don't think the statement that you linked in your revised question needs a reference, or even an explicit proof. The fact that comodule structures can be pushed forward along coalgebra homomorphisms is sufficiently clear that you should be able to get away with stating a precise claim. Similarly, I think the statement that the structure of a $G \times H$-module is equivalent to commuting $G$-module and $H$-module structures is also straightforward enough that you can state it without proof or reference. (Others may disagree, but I think I'm on the majority side in this particular case.)

A common way to deal with straightforward facts that are often used later in a paper is to write the claim as a lemma, followed by:

Proof: Omitted.

If the proof is not entirely obvious, it may help to write a sentence or two on how to get through the tricky part.

• I agree with Scott here. The fact that a $G \times H$-module structure is equivalent to commuting $G$- and $H$-module structures is standard enough that a proof isn't necessary. – Chuck Hague Nov 11 '11 at 17:47
• Thank you for the advice. Let's just hope that the referee is inclined to agree :). – Mike Crumley Nov 11 '11 at 23:29

As Vladimir mentions, the statement of your theorem is unclear. However, since you have written the $G$- and $H$-actions as tensor product actions where each acts nontrivially on one tensor factor and trivially on the other, it appears that you are making the following claim (please correct me if I'm wrong):

Any representation of $G \times H$ is isomorphic to a representation of the form $V \otimes W$, where $V$ is a representation of $G$ and $W$ is a representation of $H$, and where we let the $G$-action on $W$ and the $H$-action on $V$ be trivial.

If this is the statement you intend to make, it is false. Indeed, if we take two representations $M$, $M'$ of $G$ and two representations $N$, $N'$ of $H$, then in general the $G \times H$-module $( M \otimes N ) \oplus ( M' \otimes N' )$ will not be isomorphic to any module of the form $V \otimes W$ as described above.

• Perhaps the OP meant to consider only irreducible representations of $G\times H$? With the appropriate assumptions, every irreducible representation of $G\times H$ is obtained as an external product of irreducibles of $G$ and $H$. However, I think it's not true in general, even if you restrict to indecomposables of $G\times H$. – David Jordan Nov 10 '11 at 17:39
• I agree, I don't think it's true even in the case that the $G \times H$-representation is indecomposable. My guess is that the following statement is true: Any representation of $G \times H$ has a filtration with subquotients isomorphic to modules of the form $V \otimes W$. – Chuck Hague Nov 10 '11 at 18:07
• I think there is a great deal of confusion concerning notation (which is my fault, I apologize). I will try to reformulate the question in terms of comodules over the representing Hopf algebras of the groups. – Mike Crumley Nov 10 '11 at 22:10