# Weyl's Branching Rule for $SU(N)$-Setting

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?

## 3 Answers

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

Compact Lie groups and their representations

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

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)

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.