Branching rule for $S{L}_{n+m}(\mathbb{C})\supset S{L}_n(\mathbb{C})\times S{L}_m(\mathbb{C})$

Abstract

We study the branching problem for the pair $S{L}_{n+m}(\mathbb{C})\supset S{L}_n(\mathbb{C})\times S{L}_m(\mathbb{C})$. 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 $S{L}_{n+m}$, and $N$ is the dimension of maximal unipotent subgroup in $S{L}_{n+m}$. Let $V(\lambda )$ be the irreducible representation of $S{L}_{n+m}$ 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 $V(\lambda ){|}_{S{L}_n\times S{L}_m}$ of $V(\lambda )$ to $S{L}_n\times S{L}_m$. 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.