A collective study of algebraic combinatorics

代数组合学的集体研究

• 批准号: 09304004
• 负责人: BANNAI Eiichi
• 金额: $ 10.3万
• 依托单位: KYUSHU UNIVERSITY
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (A).
• 财政年份: 1997
• 资助国家: 日本
• 起止时间: 1997 至 2000
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-09304004/
• 关键词: Algebraic combinatorics; associationscheme; spin model; spherical design; modular form; code; finite ring; codes over finite alrelian group; modular invariant; 2次形式; Terwilliger代数; Type IIコード; 有限群の不変式環

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.

这笔赠款的主要目的是支持日本代数组合学研究的全面发展，支持参加这一领域和相关领域专题讨论会的发言者和与会者的费用。每年，我们有大约2个大型会议和一些小型研讨会。这些大型会议是：第14届（东京三鹰）、第15届（金泽）、第16届（福冈）和第17届（筑波）代数组合学学术讨论会，以及近年来每年在日本国际数学研究院（RIMS）举行的代数组合学及相关学科会议，目前日本的代数组合学活动十分活跃和成功。我们已经有了相当显着的发展，特别是在主题，如分类问题的协会计划和距离经常图;对球形设计，自旋模型，并在Terwilliger代数及其连接与代表性理论。主要的研究课题之一，在附近的主要研究者一直是研究代码的各种有限环和有限阿贝尔群，然后将这些结果的研究模形式。我们已经得到了各种结果的自对偶码和II型码在各种环。此外，我们还对SL（2，Z）中模形式环同构于多项式环的有限指数子群进行了分类。我们也开始了分数权的模形式的研究，然后我们发现了关于Γ（5）的权1/5-整数的模形式的一个有趣结果（Bannai-Koike-Munemasa-Sekiguchi）.我们目前正继续朝着这一方向努力进一步推广。首席研究员也开始研究关联方案的特征表，并试图通过将它们视为模形式的有限版本来研究对象。此外，首席研究员已经开始研究有限群的模数据以及它们的模不变量。

项目成果

伊藤達郎: "Some algebra related to P-and Q-polynomial association schemes"DIMACS : Series in Discrete Math.and Theretical Computer Science/BARG. (to appear).

Tatsuro Ito：“与 P 和 Q 多项式关联方案相关的一些代数”DIMACS：离散数学和理论计算机科学系列/BARG（即将出版）。

B.Curtin and Kazumasa Nomura: "Some formulas for spin models on distance-regular graphs" J.Combinatorial Theory(B). (to appear).

B.Curtin 和 Kazumasa Nomura：“距离正则图上自旋模型的一些公式”J.组合理论（B）。

Eiichi Bannai: "An introduction to association schemes Methods of discrete mathematics"Quad.Mat.,5,Aracne,Rome. 5. 1-70 (1999)

Eiichi Bannai：“关联方案离散数学方法简介”Quad.Mat.，5，Aracne，罗马。

Ei.Bannai, S.T.Dougherty, M.Harada and M.Oura: "Type II codes, even unimodular lattices and invariant rings"IEEE Trans.Inform.Theory. 45. 1194-1205 (1999)

Ei.Bannai、S.T.Dougherty、M.Harada 和 M.Oura：“II 型码，甚至单模格和不变环”IEEE Trans.Inform.Theory。

坂内英一(with M.Koike,A.Munemasa and Jiro Sekiguchi): "Some results on modular forms-subgroups of the modular group whose ring of modular forms is a polynomial ring"ASPM series, Math.Soc.of Japan. (to appear).

Eiichi Sakauchi（与 M.Koike、A.Munemasa 和 Jiro Sekiguchi）：“关于模形式的一些结果 - 模群的子群，其模形式的环是多项式环”ASPM 系列，Math.Soc.of Japan 出版。 ）。