# Branching rules for $SU(p,q)$

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.

• Answers completely different according as you mean finite-dimensional or unitary representations (two essentially disjoint classes). Nov 16, 2017 at 18:34
• @FrancoisZiegler Thank you fit your answer. I am looking at the finite-dimensional case and am also aware of the Book ba Mckay and Patera. However, this book treats only a limited number of the above groups (rank $\leq 8$). So does this mean that the above branching rule for general p,q is not understood? Nov 17, 2017 at 23:28
• In, this question, the groups $SU(p,q-1),SU(P,q)$ can be replaced by their complexifications, namely $SL_{n-1},SL_n$ with the smaller group viewed as the matrices in the top left hand corner of the bigger group Nov 18, 2017 at 1:47
• Regarding the unitary case, there is an old paper of T. Kobayashi, providing an explicit formula for the restriction $U(p,q)\downarrow U(p,s)\times U(q-s)$, see: projecteuclid.org/download/pdf_1/euclid.pja/1195511349 Nov 18, 2017 at 2:39

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.