Scaffolds：一个基于图的Bose-Mesner代数计算系统
Scaffolds: a graph-based system for computations in Bose-Mesner algebras
摘要：
设$X$是一个有限集，$\mathsf{Mat}_X(\mathbb{C})$表示矩阵的代数，矩阵的行和列由$X$索引，元素来自作用于具有标准基$\{ \hat{x} \mid x\in X\}$的$\mathbb{C}^X$的复数。对于一个有向图$G=(V(G),E(G))$，函数$R:[m] \rightarrow V(G)$与$r_j := R(j)$，以及一个从$G$到$\mathsf{Mat}_X(\mathbb{C})$的弧的函数$w$，我们定义“支架”$\mathsf{S}(G,R;w)$为$m$折叠张量$\widehat{φ(r_1)} \otimes \widehat{φ(r_2)} \otimes \cdots \otimes \widehat{φ(r_m)}$的从$V(G)$到$X$的所有函数$φ$的和，由$G$的所有弧$e=(a,b)$上的条目$w(e)_{φ(a),φ(b)}$的乘积缩放。支架可以用来计算有向图同态和关联方案参数，如广义交集数。它们也出现在链接不变量理论和自旋模型中。这些图是由Arnold Neumaier在20世纪80年代后期引入的，并被许多研究缔合方案的作者隐含地使用。我们重新审视结果的几个作者，改写他们的证明，这些图表和某些规则的操作（或“移动”）图。我们的目标是收集和目前，在一个统一的方式，Neumaier的原始想法扩展到张量及其使用的各种作者。有时，术语“星三角图”出现，我们在本文中称之为“脚手架”。限制的情况下，边的权重是从一个连贯的代数选择，我们探索的向量空间$\mathsf{W}((G,R); \mathbb{A})$跨越的所有脚手架上定义的根图$(G,R)$和建立一个连接到图未成年人。最后，我们提出了一个关于平面脚手架的猜想，它将关联方案的对偶性与平面图的对偶性联系起来。
英文摘要：
Let $X$ be a finite set and let $\mathsf{Mat}_X(\mathbb{C})$ denote the algebra of matrices with rows and columns indexed by $X$ and entries from the complex numbers acting on $\mathbb{C}^X$ with standard basis $\{ \hat{x} \mid x\in X\}$. For a digraph $G=(V(G),E(G))$, function $R:[m] \rightarrow V(G)$ with $r_j := R(j)$, and a function $w$ from the arcs of $G$ to $\mathsf{Mat}_X(\mathbb{C})$, we define the "scaffold" $\mathsf{S}(G,R;w)$ as the sum over all functions $φ$ from $V(G)$ to $X$ of the $m$-fold tensors $\widehat{φ(r_1)} \otimes \widehat{φ(r_2)} \otimes \cdots \otimes \widehat{φ(r_m)}$ scaled by the product of the entries $w(e)_{φ(a),φ(b)}$ over all arcs $e=(a,b)$ of $G$. Scaffolds can be used to count, among other things, digraph homomorphisms and association scheme parameters such as generalized intersection numbers. They also arise in the the theory of link invariants and spin models. These diagrams were introduced in the late 1980s by Arnold Neumaier and have been used implicitly by various authors working with association schemes. We revisit results of several authors, rephrasing their proofs in terms of these diagrams and certain rules of manipulation (or "moves") on diagrams. Our goal is to collect and present, in a uniform fashion, Neumaier's original idea extended to tensors and its used by various authors. Sometimes the term "star-triangle diagram" appears for what we, in this paper, call "scaffolds". Restricting to the case where edge weights are chosen from a coherent algebra, we explore the vector space $\mathsf{W}((G,R); \mathbb{A})$ spanned by all scaffolds defined on rooted diagram $(G,R)$ and establish a connection to graph minors. We end with a conjecture about planar scaffolds that draws a connection between association scheme duality and the duality of plane graphs.
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