Branching rule for $SL_{n+m}(\mathbb{C})\supset SL_n(\mathbb{C})\times SL_m(\mathbb{C})$

We study the branching problem for the pair $SL_{n+m}(\mathbb{C})\supset SL_n(\mathbb{C})\times SL_m(\mathbb{C})$.

We describe the corresponding branching rule in terms of a semigroup $\Sigma_{n,m}\subset\Lambda^{+}\times\mathbb{Z}^N$, where $\Lambda^{+}$ is the semigroup of dominant weights of $SL_{n+m}$, and $N$ is the dimension of maximal unipotent subgroup in $SL_{n+m}$.

Let $V(\lambda)$ be the irreducible representation of $SL_{n+m}$ with the highest weight $\lambda$.

For every dominant weight $\lambda\in\Lambda^{+}$ the set of all $\sigma\in\Sigma_{n,m}$ with dominant weight $\lambda$ parametrizes the irreducible representations in the restriction $V(\lambda)|_{SL_n\times SL_m}$ of $V(\lambda)$ to $SL_{n}\times SL_{m}$.

We describe the semigroup $\Sigma_{n,m}$ as the semigroup of integral points in some polyhedral cone and we find the inequalities defining this cone.