Branching algebras for the general linear Lie superalgebra

Abstract

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.