Branching algebras for the general linear Lie superalgebra

We develop an algebraic approach to the branching of representations of the general linear Lie superalgebra 𝔤𝔩_p|q(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 𝔤𝔩_p|q(ℂ) to a regular subalgebra isomorphic to 𝔤𝔩_r|s(ℂ)⊕𝔤𝔩_r'|s'(ℂ), 𝔤𝔩_r|s(ℂ)⊕𝔤𝔩_1(ℂ)^r'+s' or 𝔤𝔩_r|s(ℂ), with r+r'=p and s+s'=q. In the case of 𝔤𝔩_r|s(ℂ)⊕𝔤𝔩_1(ℂ)^r'+s' with s=0 or s=1 but general r, we also construct a basis for the space of 𝔤𝔩_r|s(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.