Research on association schemes and related topics

本文主要从以下三个方面进行了研究：1.尾旋模型的研究。(i)我们介绍了4-权自旋模型的概念，并证明了链接不变量是从他们（与Etsuko Bannai的联合工作）。(ii)我们给出了有限交换群特征标表上的对偶和模不变性的完全分类，并证明了自旋模型可由这些解构造（与Etsuko Bannai和F.Jaeger的合作）。2.由码的重量计数器或某些有限群的多项式不变性构造各种自守形式。(i)我们证明了Jacobi形式是由192阶的2维酉反射群的同时对角作用构造的（Shepherd-Todd列表中的第9位）（与Michio Ozeki的联合工作）。(ii)重量为4的某些Jacobi形式的显式构造（与Michio Ozeki和Shinri Minashima的联合工作）。(iii)我们确定了上面（i）中提到的多项式不变量的显式基（与Etsuko Bannal，Michio Ozeki和Yasuo Teranishi的联合工作）。(iv)我们目前正在研究有限交换群上的II型加法码（与Masaaki Harada，S. Doughnut和Manabu Oura的合作工作）。3.利用初等几何思想，如正多面体和拟正多面体的分类，对m_1=3的本原对称结合方案进行了分类。我目前正在与Attila Sali合作研究m_1=4的原始对称Q-多项式结合方案的分类。

The head researcher has done the research in the following 3 directions.1.Study of spin models. (i) We introduced the concept of 4-weight spin models, and proved that link invariants are obtained from them (joint work with Etsuko Bannai). (ii) We gave the complete classifications of the dualities and the modular invariances on the character tables of finite abelian groups, and proved that spin models are constructed from such solutions (joint work with Etsuko Bannai and F.Jaeger).2.Construction of various automorphic forms from either the weight enumerators of codes or polynomial invariants of certain finite groups.(i) We proved that Jacobi forms are constructed from the simultaneous diagonal actions of the 2-dimensional unitary reflection group of order 192 (No.9 in Shepherd-Todd's list)(joint work with Michio Ozeki).(ii) Explicit constructions of certain Jacobi forms of weight 4 (joint work with Michio Ozeki and Shinri Minashima).(iii) We determined the explicit basis of the polynomial invariants mentioned in (i) above (joint work with Etsuko Bannal, Michio Ozeki and Yasuo Teranishi).(iv) We are currently studying Type II additive codes on finite abelian groups (joint work with Masaaki Harada, S.Dougherty, and Manabu Oura).3.I classified primitive symmetric association schemes with m_1=3 by using elementary geometric ideas such as the classifications of regular polyhedrons and quasi-regular polyhedrons. I am currently working on the classification of primitive symmetric Q-polynomial association schemes with m_1=4 jointly with Attila Sali.