The Subconstituent Algebra of an Association Scheme, (Part I)
The Subconstituent Algebra of an Association Scheme, (Part I)
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DOI：
10.1023/a:1022494701663
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发表时间：
1992-12
影响因子：
0.8
通讯作者：
Paul M. Terwilliger
中科院分区：
文献类型：
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作者：
Paul M. Terwilliger
介绍了一种研究具有“多个”零交点数和/或Krein参数的交换结合方案的方法，并将该方法应用于Thep-和q-多项式方案。设Y表示任意的交换结合方案，并固定任意的顶点。我们引进了一个非交换结合半单代数T=T(X)，它的结构反映了Y的组合结构。我们称之为Y关于x的次成分代数。粗略地说，T是Y的自同构群中稳定子的中心化子代数的组合模拟。一般来说，T的结构不是由Y的交数决定的，但这些参数确实提供了一些信息。实际上，对于每个消失的交数或Krein参数，我们找到了T的生成元之间的关系，我们找出了一类结构特别简单的不可约T-模型，并说这类模型的成员是薄的。在此基础上扩展，我们说如果每个不可约的T(Y)-模对Y的每个顶点都薄。我们计算了当YisP-和Q-多项式时可能的薄的不可约T-模。具有足够大的维度的索引由四个有界整数参数来索引。如果Y是薄的，则“足够大的维”意味着“至少四维”.我们给出了薄P-多项式和Q-多项式方案的一个组合刻画，并给出了这些对象的一些例子.对于每个例子，我们展示哪些不可约T-模是实际存在的，最后我们用一些猜想和公开问题来结束。
We introduce a method for studying commutative association schemes with “many” vanishing intersection numbers and/or Krein parameters, and apply the method to theP- andQ-polynomial schemes. LetYdenote any commutative association scheme, and fix any vertexxofY. We introduce a non-commutative, associative, semi-simple-algebraT=T(x) whose structure reflects the combinatorial structure ofY. We callT the subconstituent algebra of Y with respect to x. Roughly speaking,Tis a combinatorial analog of the centralizer algebra of the stabilizer ofxin the automorphism group ofY.In general, the structure ofTis not determined by the intersection numbers ofY, but these parameters do give some information. Indeed, we find a relation among the generators ofTfor each vanishing intersection number or Krein parameter.We identify a class of irreducibleT-moduIes whose structure is especially simple, and say the members of this class arethin. Expanding on this, we sayYisthinif every irreducibleT(y)-module is thin for every vertexyofY. We compute the possible thin, irreducibleT-modules whenYisP- andQ-polynomial. The ones with sufficiently large dimension are indexed by four bounded integer parameters. IfYis assumed to be thin, then “sufficiently large dimension” means “dimension at least four”.We give a combinatorial characterization of the thinP- andQ-polynomial schemes, and supply a number of examples of these objects. For each example, we show which irreducibleT-modules actually occur.We close with some conjectures and open problems.