A basis for the symplectic group branching algebra

Abstract

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 finite-dimensional 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 × • • • × 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 explicitly described toric variety.

An analysis of the multiplicity spaces in branching of symplectic groups

“…Properties of the basis of B appearing in Corollary 3.8 are studied in [4]; in particular, we show that it is a standard monomial basis, i.e., it satisfies a straightening law. We then use that to describe an explicit toric deformation of Spec(B).…”

“…In [4], we study properties of this basis, and, in particular, show that it is a standard monomial basis, i.e. it satisfies a straightening algorithm.…”

On the SO(n + 3) to SO(n) branching multiplicity space

“…, µ d ) of the highest weight of π and σ respectively. Wallach and Yacobi [WY09] (see also [Ya10] and [KY12]) gave a clean decomposition for the multiplicity space. They proved that (1.5) Hom K (σ, π) ≃ τ (1) ⊗ · · · ⊗ τ (d+1)…”

Hibi Algebras and Representation Theory

“…Then, the multichains of B n,m,k , the corresponding GT patterns, and the Hibi algebra attached to them can be used to describe branching rules for some pairs (G, H) of classical groups, that is, how a representation of G decomposes into irreducible representations of a subgroup H of G. See [23,26,31,32].…”