琼斯对
Jones Pairs
摘要：
受琼斯的辫子群表示的启发，我们定义{ \sl a Jones pair}为一对$\nbyn$矩阵$(A,B)$，使得自同态$X_A$和$\D_B$形成辫子群的表示。当$A$和$B$是II型矩阵时，我们称$(A,B)$ { \sl为可逆琼斯对}。本文发展了琼斯对理论。我们的目的是用琼斯对的观点研究缔合方案、自旋模型和四权自旋模型之间的联系。我们使用Nomura的方法从矩阵$(A,B)$构造一对代数，我们称之为$(A,B)$的Nomura代数。这些代数成为本论文的核心工具。我们将在第\ref{Nomura}章和第\ref{IINom}章中探讨它们的性质。在第\ref{JP}章中，我们介绍了琼斯对。我们证明了四权自旋模型和可逆琼斯对的等价性。我们扩展了一些现有的概念，四重量自旋模型琼斯对。在第\ref{SpinModels}章中，我们对自旋模型下的Bose-Mesner代数的一些著名结果给出了新的证明。论文的主要结果在第\ref{InvJP}章中给出。我们证明了每一个四权自旋模型都来自于一个对称自旋模型（直到奇规范等价）。我们提出了四个玻色-梅斯纳代数与每个四权自旋模型。我们研究了这些代数之间的关系。特别是，我们提供了一个策略来搜索四个权重自旋模型。这种策略类似于Bannai，Bannai和Jaeger为寻找自旋模型而给出的方法。
英文摘要：
Motivated by Jones' braid group representations constructed from spin models, we define {\sl a Jones pair} to be a pair of $\nbyn$ matrices $(A,B)$ such that the endomorphisms $X_A$ and $\D_B$ form a representation of a braid group. When $A$ and $B$ are type-II matrices, we call $(A,B)$ {\sl an invertible Jones pair}. We develop the theory of Jones pairs in this thesis. Our aim is to study the connections among association schemes, spin models and four-weight spin models using the viewpoint of Jones pairs. We use Nomura's method to construct a pair of algebras from the matrices $(A,B)$, which we call the Nomura algebras of $(A,B)$. These algebras become the central tool in this thesis. We explore their properties in Chapters \ref{Nomura} and \ref{IINom}. In Chapter \ref{JP}, we introduce Jones pairs. We prove the equivalence of four-weight spin models and invertible Jones pairs. We extend some existing concepts for four-weight spin models to Jones pairs. In Chapter \ref{SpinModels}, we provide new proofs for some well-known results on the Bose-Mesner algebras associated with spin models. We document the main results of the thesis in Chapter \ref{InvJP}. We prove that every four-weight spin model comes from a symmetric spin model (up to odd-gauge equivalence). We present four Bose-Mesner algebras associated to each four-weight spin model. We study the relations among these algebras. In particular, we provide a strategy to search for four-weight spin models. This strategy is analogous to the method given by Bannai, Bannai and Jaeger for finding spin models.
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