标题： 一般线性李超代数的分支代数 显示英文标题

标题： Branching algebras for the general linear Lie superalgebra

摘要： 我们开发了一种代数方法，用于研究一般线性李超代数$\mathfrak{gl}_{p|q}({\mathbb C})$的表示分支，通过构造某些超级交换代数，其结构编码了分支规则。 Using this approach, we derive the branching rules for restricting any irreducible polynomial representation $V$ of $\mathfrak{gl}_{p|q}({\mathbb C})$ to a regular subalgebra isomorphic to $\mathfrak{gl}_{r|s}({\mathbb C})\oplus \mathfrak{gl}_{r'|s'}({\mathbb C})$, $\mathfrak{gl}_{r|s}({\mathbb C})\oplus\mathfrak{gl}_1({\mathbb C})^{r'+s'}$ or $\mathfrak{gl}_{r|s}({\mathbb C})$, with $r+r'=p$ and $s+s'=q$. 在$\mathfrak{gl}_{r|s}({\mathbb C})\oplus\mathfrak{gl}_1({\mathbb C})^{r'+s'}$与$s=0$或$s=1$的情况下，但一般为$r$时，我们还构造了$V$中$\mathfrak{gl}_{r|s}({\mathbb C})$高权向量空间的基；当$r=s=0$时，分支规则导致了$V$的权重重数的显式表达式，这些表达式以 Kostka 数表示。 显示英文摘要

摘要： We develop an algebraic approach to the branching of representations of the general linear Lie superalgebra $\mathfrak{gl}_{p|q}({\mathbb C})$, by constructing certain super commutative algebras whose structure encodes the branching rules. Using this approach, we derive the branching rules for restricting any irreducible polynomial representation $V$ of $\mathfrak{gl}_{p|q}({\mathbb C})$ to a regular subalgebra isomorphic to $\mathfrak{gl}_{r|s}({\mathbb C})\oplus \mathfrak{gl}_{r'|s'}({\mathbb C})$, $\mathfrak{gl}_{r|s}({\mathbb C})\oplus\mathfrak{gl}_1({\mathbb C})^{r'+s'}$ or $\mathfrak{gl}_{r|s}({\mathbb C})$, with $r+r'=p$ and $s+s'=q$. In the case of $\mathfrak{gl}_{r|s}({\mathbb C})\oplus\mathfrak{gl}_1({\mathbb C})^{r'+s'}$ with $s=0$ or $s=1$ but general $r$, we also construct a basis for the space of $\mathfrak{gl}_{r|s}({\mathbb C})$ highest weight vectors in $V$; when $r=s=0$, the branching rule leads to explicit expressions for the weight multiplicities of $V$ in terms of Kostka numbers.