Scaffolds: A graph-theoretic tool for tensor computations related to Bose-Mesner algebras
Scaffolds: A graph-theoretic tool for tensor computations related to Bose-Mesner algebras
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DOI：
10.1016/j.laa.2021.02.009
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发表时间：
2021-02
影响因子：
1.1
通讯作者：
W. Martin
中科院分区：
文献类型：
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作者：
W. Martin
在Terwilliger， Neumaier和Jaeger的早期思想的基础上，我们引入了关联方案研究中出现的某些张量的图示符号。这些张量，我们称之为“支架”，遵循一组简单的规则，这些规则推广了常见的线性代数运算，如迹、矩阵积和入口积。我们首先研究了脚手架上的一组基本“移动”，并说明了它们在组合学中的应用。接下来，我们将回顾Dickie， Suzuki和Terwilliger的研究结果。主要的新结果处理了从固定的相干代数和各种底层图中选择边权的支架向量空间之间的关系。作为一个结果，我们提供了三重正则和对偶三重正则关联格式的Terwilliger代数的简单描述。最后，我们给出了将玻色-梅斯纳代数的对偶性与圆平面图的图论对偶性联系起来的一个猜想。
We introduce a pictorial notation for certain tensors arising in the study of association schemes, based on earlier ideas of Terwilliger, Neumaier and Jaeger. These tensors, which we call “scaffolds”, obey a simple set of rules which generalize common linear-algebraic operations such as trace, matrix product and entrywise product. We first study an elementary set of “moves” on scaffolds and illustrate their use in combinatorics. Next we re-visit results of Dickie, Suzuki and Terwilliger. The main new results deal with the relationships among vector spaces of scaffolds with edge weights chosen from a fixed coherent algebra and various underlying diagrams. As one consequence, we provide simple descriptions of the Terwilliger algebras of triply regular and dually triply regular association schemes. We finish with a conjecture connecting the duality of Bose-Mesner algebras to the graph-theoretic duality of circular planar graphs.