On the Terwilliger algebra of bipartite distance-regular graphs with and

详细信息    查看全文

• 作者：Safet Penjić ^{safet.penjic@iam.upr.si}
• 关键词：Distance-regular graphs ; Terwilliger algebra ; Subconstituent algebra
• 刊名：Discrete Mathematics
• 出版年：2017
• 出版时间：March 2017
• 年：2017
• 卷：340
• 期：3
• 页码：452-466
• 全文大小：856 K
• 卷排序：340

文摘

Let ΓΓ denote a bipartite distance-regular graph with diameter D≥4D≥4 and valency k≥3k≥3. Let XX denote the vertex set of ΓΓ, and let AA denote the adjacency matrix of ΓΓ. For x∈Xx∈X and for 0≤i≤D0≤i≤D, let Γi(x)Γi(x) denote the set of vertices in XX that are distance ii from vertex xx. Define a parameter Δ2Δ2 in terms of the intersection numbers by Δ2=(k−2)(c3−1)−(c2−1)p222. It is known that Δ2=0Δ2=0 implies that D≤5D≤5 or c2∈{1,2}c2∈{1,2}.For x∈Xx∈X let T=T(x)T=T(x) denote the subalgebra of MatX(C)MatX(C) generated by A,E0∗,E1∗,…,ED∗, where for 0≤i≤D0≤i≤D, Ei∗ represents the projection onto the iith subconstituent of ΓΓ with respect to xx. We refer to TT as the Terwilliger algebra   of ΓΓ with respect to xx. By the endpoint   of an irreducible TT-module WW we mean min{i|Ei∗W≠0}.We find the structure of irreducible TT-modules of endpoint 2 for graphs ΓΓ which have the property that for 2≤i≤D−12≤i≤D−1, there exist complex scalars αiαi, βiβi such that for all x,y,z∈Xx,y,z∈X with ∂(x,y)=2,∂(x,z)=i,∂(y,z)=i, we have αi+βi|Γ1(x)∩Γ1(y)∩Γi−1(z)|=|Γi−1(x)∩Γi−1(y)∩Γ1(z)|αi+βi|Γ1(x)∩Γ1(y)∩Γi−1(z)|=|Γi−1(x)∩Γi−1(y)∩Γ1(z)|, in case when Δ2=0Δ2=0 and c2=2c2=2. The case when Δ2=0Δ2=0 and c2=1c2=1 is already studied by MacLean et al. [15].We show that if ΓΓ is not almost 2-homogeneous, then up to isomorphism there exists exactly one irreducible TT-module with endpoint 2 and it is not thin. We give a basis for this TT-module, and we give the action of AA on this basis.