Let $U(n)$ be the group of unitary $n\times n$ matrices over $\mathbb{C}$. Is there a classification of the continuous, injective group homomorphisms $U(m)\to U(n)$? If so, is there a modern account of the proof?

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dimensionsof the irreducibles for $U(m)$ with $m\ge 3$ is more complicated than the highest-weight classification, but Kostant's weight multiplicity formula tells the dimensions of the weight spaces, giving finer information. – paul garrett Dec 16 '12 at 18:31