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(1) Is it true that the category of representations of $U(n)$ is equivalent to the category of representations of $SU(N) \times U(1)$? If so, how is it proved, or what is a good reference. (I guess one uses the $N$-fold cover of $U(N)$ by $SU(N) \times U(1)$.)

(2) The representations of $U(N)$ are labelled by tuples $\alpha \in {\mathbf Z}^n$, while the representations of $SU(N) \times U(1)$ are labelled by certain tuples $\beta \in {\mathbf N}^{N-1} \times {\mathbf Z}$. Assuming (1) to be true, how do the two labellings relate to each other, ie given an $\alpha$, how can one find a $\beta$?

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    $\begingroup$ Even if such an equivalence of categories exists, it cannot respect the tensor product structure and the duality of representations. Otherwise the Tannaka-Krein theorem would imply an isomorphism of groups $U(n)$ and $SU(n)\times U(1)$, which are not isomorphic. $\endgroup$
    – asv
    Commented Jun 20, 2015 at 15:03
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    $\begingroup$ The sentence "respect the tensor product structure and the duality of representations" is equivalent to "respect the tensor product structure". There is no such thing as "respecting the duality". Or, in other words, "respecting the duality" is a formal consequence of "respecting the tensor product structure". $\endgroup$
    – André Henriques
    Commented Jun 20, 2015 at 22:05
  • $\begingroup$ There are two possible ways of interpreting the sentence "it cannot respect the tensor product structure". It matters whether or not not includes the symmetry isomorphisms. Have a look at the first 3 paragraphs of the intro of arxiv.org/pdf/math/0007196.pdf for a relevant discussion. $\endgroup$
    – André Henriques
    Commented Jun 21, 2015 at 23:00

2 Answers 2

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Any two semisimple abelian categories with countably many simple objects will be equivalent: just choose a bijection between the sets of simple objects. For example, in your question, any choice of bijection between $\mathbb Z^n$ and $\mathbb N^{n-1} \times \mathbb Z$ will give you an equivalence. More generally, the categories of representations of any two (non-discrete) compact Lie groups will be equivalent.

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  • $\begingroup$ Actually, if one only cares about an equivalence of categories, one can also twist by an automorphism of the field of complex numbers... (thus, please, before answering the question, it is better to rephrase and thus improve the question: the question should have been about a tensor equivalence) $\endgroup$
    – André Henriques
    Commented Jun 20, 2015 at 22:08
  • $\begingroup$ @André: Fair point! I started to write a more detailed answer, then remembered that I had other things to do... probably this answer should have been a comment. $\endgroup$
    – Sam Gunningham
    Commented Jun 21, 2015 at 15:02
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Regarding (2): In the context of rational complex representations of $GL_n$ vs those of $SL_n$, your extra factor is accounted for by the determinant. Given a rational representation of $GL_n$, $V$, its highest weight can be written as decreasing sequence $\lambda$. If we let $\mu =\lambda - \lambda_n(1,\dots,1)$ then $V$ is isomorphic to the tensor product of the irreducible representation with highest weight $\mu$ with $\det^{\otimes \lambda_n}$.

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