A basis for the symplectic group branching algebra

The symplectic group branching algebra, B\mathcal {B}, is a graded algebra whose components encode the multiplicities of irreducible representations of Sp_2n−2(ℂ) in each finite-dimensional irreducible representation of Sp_2n (ℂ). By describing on B\mathcal {B} an ASL structure, we construct an explicit standard monomial basis of B\mathcal {B} consisting of Sp_2n−2(ℂ) highest weight vectors. Moreover, B\mathcal {B} is known to carry a canonical action of the n-fold product SL₂×⋯×SL₂, and we show that the standard monomial basis is the unique (up to scalar) weight basis associated to this representation. Finally, using the theory of Hibi algebras we describe a deformation of Spec(B)\mathrm{Spec}(\mathcal {B}) into an explicitly described toric variety.