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 $Σ_{n,m}\subsetΛ^{+}\times\mathbb{Z}^N$, where $Λ^{+}$ is the semigroup of dominant weights of $SL_{n+m}$, and $N$ is the dimension of maximal unipotent subgroup in $SL_{n+m}$. Let $V(λ)$ be the irreducible representation of $SL_{n+m}$ with the highest weight $λ$. For every dominant weight $λ\inΛ^{+}$ the set of all $σ\inΣ_{n,m}$ with dominant weight $λ$ parametrizes the irreducible representations in the restriction $V(λ)|_{SL_n\times SL_m}$ of $V(λ)$ to $SL_{n}\times SL_{m}$. We describe the semigroup $Σ_{n,m}$ as the semigroup of integral points in some polyhedral cone and we find the inequalities defining this cone.