TITLE: Weyl's Branching Rule for $SU(N)$-Setting
QUESTION [5 upvotes]: 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?

REPLY [3 votes]: 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.