AI 中文总结
研究$SL_{n+m}(\mathbb{C})\supset SL_n(\mathbb{C})\times SL_m(\mathbb{C})$的分支问题，通过半群$\Sigma_{n,m}$描述分支规则，将其视为多面锥中的整点半群并找出定义锥的不等式，以此参数化不可约表示。
AI 中文摘要
我们研究了$SL_{n+m}(\mathbb{C})\supset SL_n(\mathbb{C})\times SL_m(\mathbb{C})$这一对的分支问题。我们用半群$\Sigma_{n,m}\subset\Lambda^{+}\times\mathbb{Z}^N$描述了相应的分支规则，其中$\Lambda^{+}$是$SL_{n+m}$的优势权半群，$N$是$SL_{n+m}$中极大幂幺子群的维数。设$V(\lambda)$是$SL_{n+m}$具有最高权$\lambda$的不可约表示。对于每个优势权$\lambda\in\Lambda^{+}$，具有优势权$\lambda$的所有$\sigma\in\Sigma_{n,m}$的集合参数化了$V(\lambda)$到$SL_{n}\times SL_{m}$的限制$V(\lambda)|_{SL_n\times SL_m}$中的不可约表示。我们将半群$\Sigma_{n,m}$描述为某个多面锥中的整点半群，并找到了定义这个锥的不等式。
英文摘要
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.
