2000 Fiscal Year Final Research Report Summary
Combinatorics in representation theory of classical groups and quantum groups, and applications
| Project/Area Number |
10440004
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| Research Category |
Grant-in-Aid for Scientific Research (B).
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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 | Nagoya University |
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Principal Investigator |
OKADA Soichi Nagoya Univ., Grad.School of Mathematics, Assoc.Prof., 大学院・多元数理科学研究科, 助教授 (20224016)
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| Co-Investigator(Kenkyū-buntansha) |
YAMADA Hiro-fumi Okayama Univ., Department of Mathematics, Prof., 理学部, 教授 (40192794)
KOIKE Kazuhiko Aoyama Gakuin Univ., Department of Mathematical, Prof., 理工学部, 教授 (70146306)
KASHIWARA Masaki Kyoto Univ., Research Institute for Mathematical Sciences, Prof., 数理解析研究所, 教授 (60027381)
TERADA Itaru Univ.of Tokyo, Grad.School of Mathematical Sciences, Assoc.Prof., 大学院・数理科学研究科, 助教授 (70180081)
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| Project Period (FY) |
1998 – 2000
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| Keywords | representations / classical groups / quantum groups / Young diagrams / Schur functions / RS correspondence / alternating sign matrix / KL conjecture |
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| Research Abstract |
In this research project, we obtained the following results.
1. Okada obtained explicit branching rules for the tensor products and restrictions of the irreducible representations of the classical groups corresponding to nearly-rectangular shaped Young diagrams. And he proved that the partition functions of the square ice model related to the alternating sign matrices with symmetry can be expressed in terms of the irreducible characters of the classical groups.
2. Kashiwara described the crystal bases for the quantum group U_q (gl(m, n))(with G.Benkart and S.Kang). Also, in the study of D-modules on the flag varieties, he proved the Kazhdan-Lusztig conjecture for the affine Lie algebras at a non-critical level (with T.Tanisaki), and showed that the duality for D-modules on the flag variety corresponds to that of Harish-Chandra modules (with D.Bartlet).
3. Koike described, in terms of generalized Brauer diagrams, the structure of the centralizer algebra of the spin groups on the tensor product of the basic spin representation and the tensor powers of the natural representation. Also he gave a realization of irreducible representations of the spin groups in the above tensor products.
4. Terada gave an geometric interpretation to the Robinson-Schensted correpondence between Brauer diagrams and up-down tableaux. And he constructed an Robinson-Schensted-type bijection for the Weil representation of sp (2n)(with T.Roby).
5. Yamada described weight vectors in the basic representations of some affine Lie algebras in terms of symmetric functions (with T.Nakajima), and found an interesting facts on the spin decomposition matrices of the symmetric groups. And He found the Littlewood's multiple formula for the spin irreducible characters of the symmetric groups (with H.Mizukawa).
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