The Terwilliger Algebra Associated with a Set of Vertices in a Distance-Regular Graph
The Terwilliger Algebra Associated with a Set of Vertices in a Distance-Regular Graph
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DOI：
10.1007/s10801-005-2504-4
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发表时间：
2005-08
影响因子：
0.8
通讯作者：
H. Suzuki
中科院分区：
文献类型：
--
作者：
H. Suzuki
令 Γ 为直径 D 的距离正则图。设 X 表示 Γ 的顶点集，并设 Y 为 X 的非空子集。我们定义一个代数 τ = τ(Y)。该代数是有限维且半简单的。如果Y 由单个顶点组成，则 τ 是 P. Terwilliger 定义的相应子成分代数。我们研究不可约 τ 模。我们将不可约 τ 模块上的端点和稀疏条件定义为 Y 包含单个顶点的情况的概括。我们确定不可约模何时变薄。当模由Y的特征向量生成时，当且仅当Y是Γ的完全正则码时，它是薄的。通过考虑合适的子集Y，端点i的每个不可约τ(x)-模可以被视为端点0的不可约τ(Y)-模。
Let Γ be a distance-regular graph of diameterD. LetXdenote the vertex set of Γ and letYbe a nonempty subset ofX. We define an algebra τ = τ(Y). This algebra is finite dimensional and semisimple. IfYconsists of a single vertex then τ is the corresponding subconstituent algebra defined by P. Terwilliger. We investigate the irreducible τ-modules. We define endpoints and thin condition on irreducible τ-modules as a generalization of the case whenYconsists of a single vertex. We determine when an irreducible module is thin. When the module is generated by the characteristic vector ofY, it is thin if and only ifYis a completely regular code of Γ. By considering a suitable subsetY, every irreducible τ(x)-module of endpointican be regarded as an irreducible τ(Y)-module of endpoint 0.