What is the branching rule for the subgroup $SU(p,q-1)\subset SU(p,q)$, i.e., the structure of the restriction of irreducible, finite-dimensional representations of $SU(p,q)$ to $SU(p,q-1)$? I would appreciate any reference and comments.



Branching rules for $SU(p,q)$
As was indicated above your question is the equivalent to the branching rules for $sl\_{n-1} \to sl\_n$. This is a well-known branching rule and it is given by the following formula: If the representation of $sl\_n$ is given by the highest weight $(\lambda\_1 \geq \cdots \geq\lambda\_n)$ then it decomposes upon restricting to $sl\_{n-1}$ to the direct sum of irreducible representations with highest weights $(\lambda^{\prime}\_1 \geq \cdots \geq\lambda^{\prime}\_{n-1})$ satisfying $\lambda\_{i+1} \leq \lambda^{\prime}\_i\leq \lambda\_{i}$. This is equivalent to removing some boxes from the corresponding Young diagramm.


