Project/Area Number  10440011 
Research Category 
GrantinAid for Scientific Research (B).

Section  一般 
Research Field 
Algebra

Research Institution  KYUSHU UNIVERSITY 
Principal Investigator 
BANNNAI Etsuko Kyushu Univ., Graduate School of Math., Assoc. Professor, 大学院・数理研究院, 助教授 (00253394)

CoInvestigator(Kenkyūbuntansha) 
NOMURA Kazumasa Tokyo Medical and Dental Univ., Faculty of Cultural Science, Professor, 教養部, 教授 (40111645)
ITO Tatsuro Kanazawa Univ., Faculty of Science, Professor, 理学部, 教授 (90015909)
MUNEMASA Akihiro Kyushu Univ., Graduate School of Math., Assoc. Professor, 大学院・数理研究院, 助教授 (50219862)
BANNAI Etsuko Kyushu Univ., Graduate School of Math., Professor, 大学院・数理研究院, 教授 (10011652)

Project Fiscal Year 
1998 – 2001

Project Status 
Completed(Fiscal Year 2001)

Budget Amount *help 
¥5,300,000 (Direct Cost : ¥5,300,000)
Fiscal Year 2001 : ¥1,400,000 (Direct Cost : ¥1,400,000)
Fiscal Year 2000 : ¥1,300,000 (Direct Cost : ¥1,300,000)
Fiscal Year 1999 : ¥1,300,000 (Direct Cost : ¥1,300,000)
Fiscal Year 1998 : ¥1,300,000 (Direct Cost : ¥1,300,000)

Keywords  Spin model / Association Schemes / designs / BoseMesnsr algebra / type II matrices / sdistance sets / modular invariance / 4weight spin models / スピンモデル / アソシエーションスキーム / デザイン / BoseMesner代数 / type II行列 / s距離集合 / モジュラー不変性 / 4weight spin model / typeII行列 / 代数的組合せ論 / self dual / symmetric design / SDP / 自己双対性 
Research Abstract 
The main purpose of this research was to find out sets of points with finite cardinality with a "good" configuration through investigation of relations between spin models and association schemes. Association scheme is the one of the most important objects in the study of algebraic combinatorics. On the other hand spin models give topological invariants of the knots and links in the 3 dimensional Euclidean space R3. During the period we received this grant we obtained the following results. (1) We found that BoseMesner algebras attached to a 4weight spin model coincide with a unique BoseMesner algebras. (2) We tried to classify 4weight spin models with small size and obtained some partial results. (3) We found out that the existence of a 4weight spin model with exactly two valued on W_2 is equivalent to the existence of a symmetric design with some polarity. (4) We defined type II codes on finite abelian groups using the solutions of modular invariance equations of finite abelian group association schemes. The solutions of this modular invariance equations are known to give spin models. (5) We found out that a upper bound for the cardinality of an sdistance set in Euclidean spaces coincides with the lower bound for the cardinality of 2sdesign given by DelsarteSeidel. In particular we found out that if we assume that sdistance set is antipodal, then a upper bound coincides with the lower bound for the cardinality of antipodal 2s  1design given by DelsarteSeidel. However the situation is different from the spherical case. The grant we received was mainly used for the travel expenses of us and also the researchers in Japan or oversea who are working on related subject with our research. We could make important discussions with many researchers in related topics. We could organized "Mini Conference on Algebraic Combinatorics" at Kyushu University 4 times.
