2004 Fiscal Year Final Research Report Summary
Algebraic Analysis of Representation Theory
| Project/Area Number |
13440006
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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 | KYOTO UNIVERSITY |
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Principal Investigator |
KASHIWARA Masaki KYOTO UNIVERSITY, Research Institute for Mathematical Sciences, Professor, 数理解析研究所, 教授 (60027381)
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| Co-Investigator(Kenkyū-buntansha) |
MIWA Tetsuji KYOTO UNIVERSITY, Graduate School of Science, Professor, 大学院・理学研究科, 教授 (10027386)
NAKAJIMA Hiraku KYOTO UNIVERSITY, Graduate School of Science, Professor, 大学院・理学研究科, 教授 (00201666)
TANIZAKI Toshiyuki Osaka City University, Faculty of Science, Professor, 大学院・理学研究科, 教授 (70142916)
内藤 聡 筑波大学, 数学系, 助教授 (60252160)
NAKASHIMA Toshiki Sophia University, Faculty of Science and Technology, Associate Professor, 理工学部, 助教授 (60243193)
ANATOL Kirillov KYOTO UNIVERSITY, Research Institute for Mathematical Sciences, Associate Professor (30346035)
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| Project Period (FY) |
2001 – 2004
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| Keywords | representation theory / crystal bases / quantum group / D-modules |
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| Research Abstract |
In this project, we focused at geometric and combinatorial aspects of representation theory. Here are main achievements in these four years.
1.(1)In the coarse of the study of the form factors of exactly solvable models as integrals, their integrands have a symmetry of affine quantum groups (Miwa et al.) The representation thus obtained is in fact the tensor product of integrable representations in positive level and negative level. This result will be proved by using the result by Nakajima given below. (2)Nakajima studied global bases and crystal bases of affine quantum groups. In particular, he showed the global bases with extremal weight correspond one-to-one to the irreducible representation of the general linear groups.
2.Schapira constructed a canonical stack on the symplectic manifolds. It contains a parameter, and when it vanishes, the stack is equivalent to the one of modules over the ring of the functions. 3.Tanisaki achieved the one-to-one correspondence between D-modules on the quantized flag manifold and the modules over the quantum group. 4.Nakashima constructed geometric crystals associated to the Schubert cells and prove that their ultra-discretization coincide with the crystal for Demazure modules. Kashiwara and Nakashima, together with Okado, is studying the method to use another flag manifold in order to obtai the perfect crystals.
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