The Terwilliger algebra of the Grassmann scheme J(N,D) revisited from the viewpoint of the quantum affine algebra Uq(sl?2)
The Terwilliger algebra of the Grassmann scheme J(N,D) revisited from the viewpoint of the quantum affine algebra Uq(sl?2)
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从量子仿射代数 Uq(sl?2) 的角度重温格拉斯曼方案 J(N,D) 的特威利格代数
DOI：
10.1016/j.laa.2020.03.005
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发表时间：
2020
影响因子：
1.1
通讯作者：
Watanabe Yuta
中科院分区：
文献类型：
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作者：
Liang Xiaoye;Ito Tatsuro;Watanabe Yuta
设Δ是满足0≤ D-d2 ≤ ν≤ μ≤ D-d ≤ D的整数的所有四元组（ν，μ，d，e）的集合，e+ d+ D是偶数，|e| ≤ 2 ν− D+ d，d∈{e+ D− 2 ν，min <${D− μ，e+ D− 2 ν+ 2（N− 2 D）}}。文[6]证明了Grassmann概型J q（N，D），N≥ 2D的Terwilliger代数T的不可约T-模W的同构类由其端点ν，对偶端点μ，直径d和辅助参数e决定，这些端点v，对偶端点μ，直径d和辅助参数e来自于W的伦纳德系，并且没有证明地声称四重模W的同构类是由它们的端点v，对偶端点μ，直径d和辅助参数e决定的当d≥ 1时，（v，μ，d，e）属于Δ.设Λ是满足0≤ α≤ D− ρ 2，0≤ β≤ N− D− ρ 2，0≤ α+ β≤ D− ρ的非负整数三元组（α，β，ρ）的集合。我们构造了一个从Λ到Δ的映射，当N> 2D时它是双射的，当N= 2D时它是2：1的.通过将J q（N，D）的标准模嵌入到允许U q（s l <$2）-模结构的更大空间中，我们证明了集合Λ自然地参数化了不可约T-模的同构类[9]。作为副产品，我们有以下内容：对于固定的ρ，0≤ ρ≤ D，设置N′= N− 2 ρ，D′= D− ρ，并且Λ ρ={（α，β）|（α，β，ρ）∈ Λ}。则Λ ρ恰好是约翰逊概型J（N′，D′）[3]的不可约T-模的同构类的参数化集.
Let Δ be the set of all quadruples (ν, μ, d, e) of integers that satisfy 0≤ D− d 2≤ ν≤ μ≤ D− d≤ D, e+ d+ D is even,| e|≤ 2 ν− D+ d, d∈{e+ D− 2 ν, min{D− μ, e+ D− 2 ν+ 2 (N− 2 D)}}. In [6], it is shown for the Terwilliger algebra T of the Grassmann scheme J q (N, D), N≥ 2 D, 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 satisfy 0≤ α≤ D− ρ 2, 0≤ β≤ N− D− ρ 2, 0≤ α+ β≤ D− ρ. We construct a mapping from Λ to Δ which is bijective if N> 2 D and 2: 1 if N= 2 D. We show that the set Λ naturally parameterizes the isomorphism classes of irreducible T-modules, by embedding the standard module of J q (N, D) in a bigger space that allows a U q (s l ˆ 2)-module structure [9]. As a byproduct we have the following: for a fixed ρ, 0≤ ρ≤ D, set N′= N− 2 ρ, D′= D− ρ, and Λ ρ={(α, β)|(α, β, ρ)∈ Λ}. Then Λ ρ is precisely the set that parameterizes the isomorphism classes of irreducible T-modules for the Johnson scheme J (N′, D′)[3].