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What is known about the subgroup $U(n)\subset U(mn)$ for $m,n\in\mathbb{N}$ given by the diagonal embedding $$ \alpha\mapsto \text{diag}(\alpha,\cdots, \alpha),$$ for $\alpha$ appearing $m$ times? For example, for $n=1$, this is a central subgroup of $U(m)$, the diagonal matrices with $U(1)$ entries, and this gives rise to the sequence $$ 1\to U(1)\to U(m)\to U(m)/U(1)=PU(m)\to 1. $$ For $n\geq 2$ the subgroup is obviously non-abelian, but is it normal? If not, is the set of equivalence classes $U(mn)/U(n)$ anything well-known (e.g. just as $U(m+1)/U(m)=S^{2n+1}$)?

For what is worth, something remotely close to this setting seems to go under the name of colligations, as described for instance in many papers by Yu. A. Neretin.

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    $\begingroup$ $U(m)$ has almost no normal subgroups; more precisely I think every nonabelian normal subgroup contains $SU(m)$. $\endgroup$ Commented Jul 17, 2024 at 23:08
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    $\begingroup$ Indeed, $PSU(m)$ is simple even as an abstract group without taking account of the topology. So every normal subgroup of $U(m)$ is either contained in the scalars or contains $SU(m)$. $\endgroup$ Commented Jul 18, 2024 at 6:58
  • $\begingroup$ The quotient should be something like the space of ways to write $\mathbb{C}^{mn}$ as a tensor product $V \otimes \mathbb{C}^m$ but it seems a little tricky to get the details right. $\endgroup$ Commented Jul 18, 2024 at 18:14

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I claim when $m,n>1$ the subgroup $U(n)\subset U(mn)$ is never normal. The image of $\alpha=\mathrm{diag}(z,1,\dots,1)$ with $|z|=1$ is $$\mathrm{diag}(z,1^{n-1},z,1^{n-1},\dots,z,1^{n-1}),$$ where I use the superscript to denote the multiplicity. But conjugating by $$\begin{pmatrix} &1\\ 1&\\ &&I_{mn-2} \end{pmatrix}\in U(mn)$$ takes the image of $\alpha$ to $$\mathrm{diag}(1,z,1^{n-2},z,1^{n-1},\dots,z,1^{n-1}),$$ which is not in $U(n)\subset U(mn)$.

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  • $\begingroup$ Thanks for the quick response. Is there however anything that can be said about the set of conjugacy classes $U(mn)/U(n)$? Is it some known homogeneous space? $\endgroup$ Commented Jul 18, 2024 at 16:10

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