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Suppose $G$ and $H$ are discrete groups. If $\rho_G$ and $\rho_H$ are reps of $G$ and $H$ on $V_G$ and $V_H$, respectively, then we get a rep of $G\times H$ on $V_G\otimes V_H$ by sending $(g,h)$ to $\rho_G(g)\otimes\rho_H(h)$. This induces a map $$ R(G) \otimes R(H) \to R(G\times H) $$ where $R(\cdot)$ denotes the representation ring. I know, by character theory, that this is an isomorphism for $G$ and $H$ finite.

I am guessing this result is not true in general-- what is a simple counterexample?

Are there nice conditions on $G$ and $H$ under which the map above is an isomorphism?


EDIT

Because of some of the issues with the correct definition of representations rings, I will rephrase my question to precisely what I want to know:

Is it always the case that any finite dimensional representation (over $\mathbb C$) of $G \times H$ is isomorphic to $\bigoplus_j V_j \otimes W_j$ where $V_j$ is a rep of $G$ and $W_j$ is a rep of $H$?

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This might depend on how you define $R(G)$ without semisimplicity. The main result (that the tensor product of an irrep of $G$ with an irrep of $H$ is always an irrep of $G\times H$, and that every irrep of $G\times H$ can be written in this form) does hold for any groups $G$ and $H$, even infinite ones. See artofproblemsolving.com/Forum/viewtopic.php?f=61&t=418805 . –  darij grinberg Jun 7 '13 at 15:03
    
@darijgrinberg I want $R(G)$ to be the ring of all representations of $G$ (not e.g. just the ring generated by irreps). Thanks for the link-- I did not know that that result holds in general. –  Eric O. Korman Jun 7 '13 at 15:21
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@Eric: darij's point is that there are two ways to define $R(G)$ even as an additive group in the absence of semisimplicity; you can either impose $[A] = [B] + [C]$ whenever $A \cong B \oplus C$ or you can impose $[A] = [B] + [C]$ whenever there is a short exact sequence $0 \to B \to A \to C \to 0$ (the split Grothendieck group and the Grothendieck group respectively). The first option gives you a bigger group in general. –  Qiaochu Yuan Jun 7 '13 at 18:46
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Silly question, is it true for the analog of my example? Is every finite dimensional $\mathbb{C}[x,y]$ module of the form $\bigoplus_j V_j \otimes W_j$ for finite dimensional $\mathbb{C}[x]$ (or y) modules $V_j,W_j$? I kind of thought they were weirder than that –  Jack Schmidt Jun 7 '13 at 20:40
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(Take $G=H=\mathbb{Z}$ and handle inverses, which I wouldn't expect to change too much.) –  Jack Schmidt Jun 7 '13 at 20:41
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up vote 1 down vote accepted

I asked this on mathoverflow and got a counterexample: http://mathoverflow.net/questions/133409/are-all-representations-of-g-times-h-induced-from-representations-of-g-and-h .

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The same counterexample as mentioned here :P –  Jack Schmidt Jun 12 '13 at 2:37
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