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On the Wikipedia page for restricted representations

https://en.wikipedia.org/wiki/Restricted_representation

there is presented a number of explicit "branching rules". In particular, there is the Weyl's branching rule from U(N) to U(N-1) given in terms of signatures $f_1 \geq \cdots \geq f_N$, for $f_i \in \mathbb{N}$, labelling irreps of U(N). I would guess that this generalises directly to the case of branching from $SU(N)$ to $SU(N-1)$ but cannot find a reference. Can someone suggest a reference?

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The question is answered on page 385 of the classical Zhelobenko book

Compact Lie groups and their representations

for the more general case of $SU(n+m)/SU(n) \times SU(m)$.

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Maybe the following paper might prove helpful to your question:

Masatoshi Yamazaki, Branching Diagram for Special Unitary Group SU(n), J. Phys. Soc. Jpn. 21, pp. 1829-1832 (1966)

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Every irrep of $SU(n)$ extends to irreps of $U(n)$, and conversely, the restriction of any irrep of $U(n)$ to $SU(n)$ remains irreducible. If your dominant weight of $SU(n)$ is $(a_1,\ldots,a_{n-1})$ then extend it to $(\sum_{i=1}^{n-1} a_i, \sum_{i=2}^{n-1} a_i, \ldots, a_{n-1}, 0)$, apply the $U(n)$ restriction, take differences $f_i-f_{i+1}$ of the resulting signatures.

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