A Basis for the Symplectic Group Branching Algebra

TL;DR

This paper constructs an explicit standard monomial basis for the symplectic group branching algebra, revealing its algebraic structure, symmetries, and a deformation into a toric variety, advancing understanding of representation multiplicities.

Contribution

It introduces an ASL structure on the symplectic branching algebra, providing an explicit basis and connecting it to toric degenerations, which is a novel approach.

Findings

Constructed an explicit standard monomial basis for B.

Demonstrated B's canonical SL(2)^n action and basis uniqueness.

Described a deformation of Spec(B) into a toric variety.

Abstract

The symplectic group branching algebra, B, is a graded algebra whose components encode the multiplicities of irreducible representations of Sp(2n-2,C) in each irreducible representation of Sp(2n,C). By describing on B an ASL structure, we construct an explicit standard monomial basis of B consisting of Sp(2n-2,C) highest weight vectors. Moreover, B is known to carry a canonical action of the n-fold product SL(2) \times ... \times SL(2), 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) into an explicit toric variety.