2000 Fiscal Year Final Research Report Summary
Discrete and Combinatorial Geometry of finite Groups
| Project/Area Number |
11640018
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
Algebra
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| Research Institution | Yamanashi University |
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Principal Investigator |
MIYAMOTO Izumi Yamanashi University Faculty of Engineering Professor, 工学部, 教授 (60126654)
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| Co-Investigator(Kenkyū-buntansha) |
SUZUKI Tomohiro Yamanashi University Faculty of Engineering Research Assistant, 工学部, 助手 (70235977)
SATOU Masahisa Yamanashi University Faculty of Engineering Professor, 工学部, 教授 (30143952)
KURIHARA Mitsunobu Yamanashi University Faculty of Engineering Professor, 工学部, 教授 (50027372)
HANAKO Akihide Shinshu University Faculty of Science Associate Professor, 理学部, 助教授 (50262647)
NAKAI Yoshinobu Yamanashi University Faculty of Education and Human Science Professor, 教育人間科学部, 教授 (40022652)
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| Project Period (FY) |
1999 – 2000
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| Keywords | association scheme / permutation group / algebraic computation |
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| Research Abstract |
An association scheme is a discrete combinatorial geometry. Let X be a transitive permutation group on a set X.Then the orbits of G on X×X defines an association scheme. In the present research we studied association schemes. We classified the isomorphism classes of association schemes of order up to 28 as a joint work with A.Hanaki, one of the research investigator. We used computers. In oeder to construct association schemes we used a program written by C and for computing isomorphisms between association schemes we used a program written by GAP-language. Most of the obtained association schemes can be said given by groups. There are a number of exceptions but almost all of them have small ranks which correspond to the number of the orbits on X×X in group case, and they can be said to be contained in a small number of kinds. Regular groups are permutation representations as an association scheme. They are called thin. There are a classes called quasi-thin. Our classification found an example not given by a group belonging to quasi-thin class. This seems to be a hint for future research. We studied an application of our program computing isomorphisms. If an association scheme is defined by a group, then isomorphisms to itself contain the normalizer of the group. There are a couple of transitive groups of rather small degree of which normalizers are hard to compute. We applied our program to reduce the searching space of backtrack algorithm in the normalizer comutation and we have been abel to compute such normalizers within several seconds. We used an algebraic technique in the program and it was particularly effective for groups with many orbits on X×X.We are now studying the program theoretically.
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