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Consider the group $U(d_1) \otimes U(d_2)$ of "local unitary" operators acting on the complex space $\mathbb{C}^{d_1} \otimes \mathbb{C}^{d_2}$ (i.e., $U(d_1) \otimes U(d_2)$ is the group of unitary operators of the form $U \otimes V$, where $U$ and $V$ are themselves unitary operators acting on $\mathbb{C}^{d_1}$ and $\mathbb{C}^{d_2}$, respectively).

It is known (see "D. Z. Djokovic and C. K. Li, Overgroups of some classical linear groups with application to some linear preserver problems, Linear Algebra Appl. 197–198 (1994) 31–62" or "C. K. Li and N. K. Tsing, Linear operators preserving certain functions on singular values of matrices, Linear and Multilinear Algebra 26 (1990) 119–132") that only compact groups containing $U(d_1) \otimes U(d_2)$ are:

  • The full unitary group $U(d_1 d_2)$;

  • $U(d_1) \otimes U(d_2)$ itself; and

  • (if $d_1 = d_2$) the group generated by $U(d_1) \otimes U(d_2)$ and the "swap operator" $S$, defined on $\mathbb{C}^{d_1} \otimes \mathbb{C}^{d_2}$ by $S(\mathbf{a} \otimes \mathbf{b}) = \mathbf{b} \otimes \mathbf{a}$.

The above three groups are the "obvious" groups between $U(d_1) \otimes U(d_2)$ and $U(d_1d_2)$ --- the nice thing is that it turns out that there are no others.

I am interested in the natural generalization of this problem to the case where we take the tensor product of more than 2 unitary groups. My question is:

Let $G$ be a compact group of operators acting on $\mathbb{C}^{d_1} \otimes \mathbb{C}^{d_2} \otimes \cdots \otimes \mathbb{C}^{d_p}$ that contains $U(d_1) \otimes U(d_2) \otimes \cdots \otimes U(d_p)$. Is $G$ necessarily one of the "obvious" groups between $U(d_1) \otimes U(d_2) \otimes \cdots \otimes U(d_p)$ and $U(d_1d_2 \cdots d_p)$?

As noted above, the answer to this question is "yes" when $p = 2$ (and is trivially true when $p = 1$). The "obvious" groups that I'm referring to in the question above are those that arise from collecting together different tensor factors and potentially inserting swap operators where the dimensions match up. So, for example, $U(d_1 d_2) \otimes U(d_3) \otimes \cdots \otimes U(d_p)$ is one of the "obvious" groups.

To help clarify things a bit, in the $p = 3$ case, here are all of the "obvious" groups:

  • $U(d_1) \otimes U(d_2) \otimes U(d_3)$;

  • $U(d_1d_2) \otimes U(d_3)$;

  • $U(d_1) \otimes U(d_2d_3)$;

  • $U(d_1d_3) \otimes U(d_2)$ (here we understand that by $U(d_1d_3)$ I mean the unitaries acting on the first and third tensor factors, not the first and second tensor factors);

  • $U(d_1d_2d_3)$;

  • (if $d_1d_2 = d_3$) the group generated by $U(d_1d_2) \otimes U(d_3)$ and the "swap operator" $S$ defined on $(\mathbb{C}^{d_1} \otimes \mathbb{C}^{d_2}) \otimes \mathbb{C}^{d_3}$ by $S(\mathbf{a} \otimes \mathbf{b}) = \mathbf{b} \otimes \mathbf{a}$ (where $\mathbf{a} \in \mathbb{C}^{d_1} \otimes \mathbb{C}^{d_2}$ and $\mathbf{b} \in \mathbb{C}^{d_3}$); and

  • if $d_2d_3 = d_1$ then there is another similar group involving a swap operator, also for $d_1d_3 = d_2$, $d_1 = d_2$, $d_2 = d_3$, and so on.

Then the question is: are there other compact groups containing $U(d_1) \otimes U(d_2) \otimes U(d_3)$?

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  • $\begingroup$ Could you please add at least one top-level tag (those with two-letter prefix corresponding to arXiv subject classes)? Thanks in advance! $\endgroup$
    – user9072
    Feb 24, 2014 at 19:05

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