External tensor product of irreducible representations is not irreducible?ip ebe tiie 40 mpl908a Fjab653 Gr<24 0;
I'm writing up some notes, and I realize I don't have a counterexample for something I suspect is false.
Dubious claim: If $(\\pi, V)$ and $(\\rho, W)$ are irreducible representations of two groups $G$ and $H$, respectively, then the "external" tensor product $\\pi \\boxtimes \\rho$ is an irreducible representation of $G \\times H$.
Of course this is true and well-known in the usual cases, e.g., when $G$ and $H$ are finite groups. The proof I know uses the converse to Schur's Lemma, or something similar.
Is there a nice counterexample for complex representations of some infinite groups? Published somewhere?
1 Answer
You can generate examples from standard counterexamples to (generalisations of) Schur's lemma.
Let E/F be a field extension. Let $G=H=E^\\times$, acting on the F-vector space E. Then the external tensor product is not irreducible, for example the kernel of the multiplication map $E\\otimes_F E\\to E$ is a submodule.
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$\\begingroup$ I think this works. So in the context of complex representations, let $G = H = {\\mathbb C}(T)^\\times$ acting on the complex vector space $V = {\\mathbb C}(T)$. It's kind of interesting to me to see if there's a countable-dimension example (over the complex numbers), but I won't move the goalposts here! $\\endgroup$ – Marty 4 hours ago
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1$\\begingroup$ There won't be an example of countable dimension over C. Over any field, an irrep always has endomorphism ring a division algebra. There are no division algebras of countable dimension over C. Thus for countable dimension, you have Schur's Lemma so your proof of irreducibility of the tensor product should go through. $\\endgroup$ – Peter McNamara 4 hours ago
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$\\begingroup$ Well, I know the first part well. But I use the converse of Schur's Lemma to prove irreducibility of the tensor product. One can prove $End_{G \\times H}(\\pi \\boxtimes \\rho$ is ${\\mathbb C}$ using Schur's Lemma... but then what? Or am I missing something easy here? $\\endgroup$ – Marty 3 hours ago
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1$\\begingroup$ If your representations (one is enough) are finite-dimensional over a ground field it works in the same conditions as for finite groups (free is the field is algebraically closed, needs extra assumptions else). Anyway, as PeterMcNamara's answer tells, it is much more a question of field theory rather than a one of representation theory. $\\endgroup$ – Aurélien Djament 2 hours ago
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$\\begingroup$ @Marty, you can use Jacobson density theorem on V and W to prove V\\otimes W is irreducible (this approach also has the benefit of working in categories which are not semisimple). Write to me if you want more details. $\\endgroup$ – Peter McNamara 2 hours ago
