TITLE: Branching rules for $SU(p,q)$
QUESTION [6 upvotes]: 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.

REPLY [5 votes]: 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.