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 ${\mathfrak{gl}}_{r|s}(\mathbb{C})\oplus {\mathfrak{gl}}_{r^{\prime }|s^{\prime }}(\mathbb{C})$, ${\mathfrak{gl}}_{r|s}(\mathbb{C})\oplus {\mathfrak{gl}}_1(\mathbb{C}{)}^{r^{\prime }+s^{\prime }}$ or ${\mathfrak{gl}}_{r|s}(\mathbb{C})$, with $r+r^{\prime }=p$ and $s+s^{\prime }=q$. In the case of ${\mathfrak{gl}}_{r|s}(\mathbb{C})\oplus {\mathfrak{gl}}_1(\mathbb{C}{)}^{r^{\prime }+s^{\prime }}$ 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.

35 pages