1995 Fiscal Year Final Research Report Summary
Co-operative Research in Representation Theory
| Project/Area Number |
05302001
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| Research Category |
Grant-in-Aid for Co-operative Research (A)
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| Allocation Type | Single-year Grants |
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| Research Field |
Algebra
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| Research Institution | OSAKA UNIVERSITY |
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Principal Investigator |
KAWANAKA Noriaki OSAKA UNIV.FACULTY OF SCIENCE PROFESSOR, 理学部, 教授 (10028219)
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| Co-Investigator(Kenkyū-buntansha) |
YOSHIDA Tomoyuki HOKKAIDO UNIV.FACULTY OF SCIENCE PROFESSOR, 理学部, 教授 (30002265)
BANNAI Eiichi KYUSHU UNIV., GRADUATE SCHOOL OF MATH.PROFESSOR, 数理学専攻, 教授 (10011652)
MIWA Tetsuji KYOTO UNIV.RESERCH INST.MATH.SCIENCE PROFESSOR, 数理解析研究所, 教授 (10027386)
TANISAKI Toshiyuki HIROSHIMA UNIV.FACULTY OF SCIENCE PROFESSOR, 理学部, 教授 (70142916)
TAKEUCHI Mitsuhiro UNIV.OF TSUKUBA INSTITUTE OF MATH.PROFESSOR, 数学系, 教授 (00015950)
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| Project Period (FY) |
1993 – 1995
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| Keywords | Kac-Moody Lie Algebra / Quantum Group / Association Scheme / Finite Group / Unitary Refrection Group |
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| Research Abstract |
Toshiyuki Tanisaki, with Masaaki Kashiwara, succeeded in proving the Kazhdan-Lusztig type conjecture for the characters of irreducible highest weight representations of Kac-Moody Lie algebras. This completed the so-called Lusztig program, which intended to prove Kazhdan-Lusztig type conjectures for Kac-Moody Lie algebras, semisimple algebraic groups, and quantum groups whose parameters are roots of unity, simultaneously. (The equivalence of these three conjectures has already been proved by other people.) Eiichi Bannai, with Etsuko Bannai, proved the modular invariance property of the fusion algebras at algebraic level constructed from Hamming association scheme. Bannai, with Michio Ozeki, also showed that one can get modular forms by putting Jacobi theta functions into invariant polynomials of certain finite unitary reflection groups. These investigations of Bannai and his collaborators are opening a new reserch area of algebraic combinatorics. Mitsuhiro Takeuchi introduced the notion of q-representations of quantum groups, and studied q-representations of quantum special linear groups. Takeuchi also described all the automorphisms of the quantum general linear groups, and all the automorphisms and endomorphisms of quantum special linear groups. Tomoyuki Yoshida studied the number of endomorhisms between finite groups. One of his results says that the number of endomorphisms from a finite abelian group A to a finite group G is divisible by the greatest common divisors of the orders of A and G.Many of other remarkable results are reported in the Proceedings of the 12th Symposium in Algebraic Combinatorics held at Tokyo University in July, 1995.
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