The Terwilliger algebra of the Grassmann scheme Jq(N,D) revisited from the viewpoint of the quantum affine algebra [formula omitted

Abstract

Let Δ be the set of all quadruples (ν,μ,d,e) of integers that satisfy0≤D−d2≤ν≤μ≤D−d≤D,e+d+Dis even,|e|≤2ν−D+d,d∈{e+D−2ν,min⁡{D−μ,e+D−2ν+2(N−2D)}}. In [6], it is shown for the Terwilliger algebra T of the Grassmann scheme Jq(N,D), N≥2D, that the isomorphism classes of irreducible T-modules W are determined by their endpoint ν, dual endpoint μ, diameter d, and auxiliary parameter e, that come from the Leonard system attached to W, and it is claimed without proof that the quadruples (ν,μ,d,e) belong to Δ, if d≥1.Let Λ be the set of triples (α,β,ρ) of non-negative integers that satisfy0≤α≤D−ρ2,0≤β≤N−D−ρ2,0≤α+β≤D−ρ. We construct a mapping from Λ to Δ which is bijective if N>2D and 2:1 if N=2D.We show that the set Λ naturally parameterizes the isomorphism classes of irreducible T-modules, by embedding the standard module of Jq(N,D) in a bigger space that allows a Uq(slˆ2)-module structure [9].As a byproduct we have the following: for a fixed ρ, 0≤ρ≤D, set N′=N−2ρ, D′=D−ρ, and Λρ={(α,β)|(α,β,ρ)∈Λ}. Then Λρ is precisely the set that parameterizes the isomorphism classes of irreducible T-modules for the Johnson scheme J(N′,D′)[3].