A collective study of algebraic combinatorics

这笔赠款的主要目的是支持日本代数组合学研究的整体发展，支持出席这一领域和相关领域研讨会的演讲者和参与者的费用。每年，我们有大约两次大型会议和一些较小的研讨会。这些大型会议是：第14届(三中，东京)、第15届(金泽)、第16届(福冈)和第17届(筑波)代数组合学研讨会；以及过去几年在RIMS每年举行的代数组合学和/或相关子机会议。目前，日本的代数组合学活动非常活跃和成功。特别是在结合方案和距离正则图的分类问题、球面设计、自旋模型和Terweiger代数及其与表示论的联系等方面，我们取得了相当显著的进展。与主要研究者密切相关的主要研究课题之一是研究各种有限环和有限交换群上的码，并将这些结果应用到模形式的研究中。我们在不同的环上得到了关于自对偶码和II型码的各种结果。此外，我们还对模型环同构于多项式环的SL(2，Z)的有限指标子群进行了分类。我们也开始了分数权的模形式的研究，然后我们发现了一个有趣的结果(Bannai-Koike-Munemasa-Sekiguchi)关于Γ(5)的1/5-整数的模形式。我们目前正在继续朝着这个方向进行进一步的概括。首席研究人员还开始研究结合方案的特征表，并试图通过将它们视为模形式的有限版本来观察对象。同时，主要研究者也开始研究有限群的模数据以及它们的模不变量。

The main purpose of this grant was to support the overall developments of the research in algebraic combinatorics in Japan, by supporting the expenses of speakers and participants who attended the symposiums in this and related areas. Each year, we have about 2 large meetings and some smaller workshops. These large meetings were : the 14th (Mitaka, Tokyo), 15th (Kanazawa), 16th (Fukuoka) and 17 th (Tsukuba) Algebraic Combinatorics Symposiums ; and the conferences on algebraic combinatorics and/or related subjets which were held in RIMS annually for the last several years.The current activities of algebraic combinatorics in Japan is very active and successful. We have had quite notable developments, in particular, on the subjects such as classification problems of association schemes and distance-regular graphs ; on spherical designs, on spin models, and on Terwilliger algebras and its connections with representation theory. One of the main research subjects in the close neighborhood of the principal investigator has been the study of codes over various finite rings and finite abelian groups, and then to apply these results to the studies of modular forms. We have obtained various results on self-dual codes and Type II codes over various rings. In addition, we have classified the finite index subgroups of SL (2, Z) whose ring of modular forms is isomorhpic to a polynomial ring. We have also started the study of modular forms of fractional weights, and then we found an interesting result (Bannai-Koike-Munemasa-Sekiguchi) on the modular forms of weghts 1/5-integers of Γ (5). We are currently continuing to work on further generalizations in this direction. The principal investigator has also started to work on the character tables of association schemes and trying to look at the object, by looking at them as a finite version of modular forms. Also, the principal investigator has started to study the modular data of finite groups as well as their modular invariants.