Study of association schemes

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The conceptof spin model was introduced by V.F.R.Jones in the paper published in Pac.J.Math. (1989). Each spin model gives a link invariant. The main research of the principal investigator in the last one year was in the following 2 directions. (1) Generalizing the concept of spin model and find such new spin models giving link invariants, (2) To study further spin models in connection with association schemes and releted object in algebraic combinatorics, then to study the construction and classfication problems of spin models in that framework. Explicit results are as follows. (1) In a joint work with Etsuko Bannai, we intriduced the concept of generalized generalized spin models (4-weighs spin models) by further generalizing the concept of generalized spin models introduced by Kawagoe-Munemasa-Watatani. We also constructed spin models on finitte cyclic groups by using the classification of the modular invariance properties on finite cyclic groups. (2) In a jouint work with Etsuko Bannai and F.Jaeger, we proved that the modular invariance holds if a generalized spin model generates the Bose-Mesner algebra of an association scheme, and we classified the modular invariance properties on finite abelian groups. We also proved that each such solution of the modular invariance gives a spin model on the abelian group, which generalizes the work of Kac and Wakimoto of the construction of spin models on abelian groups from an even Q-form. We also obtained the classification of small spin models in a joint work with F.Jaeger and A.Sali.