The construction of four-weight spin models by using Hadamard matrices and M-structure
The construction of four-weight spin models by using Hadamard matrices and M-structure
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利用Hadamard矩阵和M结构构建四权重自旋模型
DOI：
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发表时间：
1994
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通讯作者：
M. Yamada
中科院分区：
文献类型：
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作者：
M. Yamada
尾旋模型的概念是由V.F. 1989年的琼斯K.川越，A.宗正和Y. Watatani通过去除对称性条件来推广它。最近E. Bannai和E. Bannai进一步推广了自旋模型的概念，给出了四重自旋模型或广义自旋模型。在此之前，F。耶格首先指出了自旋模型和关联方案之间的关系。K.野村从4 n阶的阿达玛矩阵构造了一个琼斯型的循环变量4fo的对称自旋模型族。V. G. Kac和M. Wakimoto证明了Jones型自旋模型和四权自旋模型可以用李代数来构造。最近K。Nomura证明了每一个对称四权自旋模型都是由一个Jones型对称自旋模型通过一个扭积构造而来。本文证明了由Jones引入的Jones型对称自旋模型可以由一个四权自旋模型构造，使得四个函数中的两个（不一定全部）是对称的.另一方面，众所周知，两个四权自旋模型的张量积也是一个四权自旋模型。我们给出了一个四权自旋模型的构造，它不是张量积构造。也就是说，如果存在一个满足一定条件的循环变量D的四权自旋模型，我们可以从它构造一个也满足相同条件的循环变量2D的四权自旋模型。我们给出了一个例子，一个四重自旋模型满足这个条件，从阿达玛矩阵和复杂的阿达玛矩阵构造。这意味着存在一个无限族的四权自旋模型。我们使用M-结构证明了这些结果。* 这项工作得到了教育、科学和文化部一般科学研究补助金的部分支持。Australasian Journal of Combinatorics IQ（1994），pp. 237-244
The concept of spin models was introduced by V.F. Jones in 1989. K. Kawagoe, A. Munemasa and Y. Watatani generalized it by removing the condition of symmetry. Recently E. Bannai and E. Bannai further generalized the concept of spin models, to give four-weight spin models or generalized spin models. Before this, F. Jaeger first pointed out the relation between spin models and association schemes. K. Nomura constructed a family of symmetric spin models of Jones type of loop variable 4fo from Hadamard matrices of order 4n. V. G. Kac and M. Wakimoto showed that spin models of Jones type and 4-weight spin models can be constructed by using Lie algebras. Recently K. Nomura proved that every symmetric four-weight spin model comes from a symmetric spin model of Jones type by a twisting product construction. In this paper, we prove that a symmetric spin model of Jones type, which was introduced by Jones, can be constructed from a four-weight spin model such that two of the four functions (not necessarily all) are symmetric. On the other hand, it is well known that the tensor product of two four-weight spin models is also a four-weight spin model. We give a construction of a four-weight spin model, which is not the tensor product construction. Namely if there exists a four-weight spin model of loop variable D satisfying a certain condition, we can construct a four-weight spin model of loop variable 2D from it, which also satisfies the same condition. We give an example of a four-weight spin model satisfying this condition, constructed from Hadamard matrices and complex Hadamard matrices. It means that there exists an infinite family of four-weight spin models. We prove these results by using an M-structure. *This work was supported in part by a Grant-in-Aid for General Scientific Research from the Ministry of Education, Science and Culture. Australasian Journal of Combinatorics IQ( 1994), pp. 237-244