Are there some natural homomorphisms between unitary groups of different dimensions, in particular surjective homomorphisms from $U(mn)$ to $U(m)$? I only know $\det : U(n) \to U(1)$ as examples.
2026-02-22 17:32:04.1771781524
homomorphism between unitary groups
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There aren't many interesting homomorphisms from $U(n)$ since it contains the subgroup $SU(n)$ of codimension $1$, which is simple modulo its finite centre. So a Lie group homomorphism from $U(n)$ must have image of dimensions $0$, $1$, $n^2-1$ or $n^2$. A homomorphism from $U(mn)$ to $U(m)$ for $m>1$ must have an image of dimension $\le1$, so is not surjective.