The construction of four-weight spin models by using Hadamard matrices and M-structure

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