2011 Fiscal Year Final Research Report
Schubert calculus on flag varieties and its application
| Project/Area Number |
21540104
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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 |
Geometry
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| Research Institution | Kagawa National College of Technology |
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Principal Investigator |
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| Co-Investigator(Renkei-kenkyūsha) |
MIMURA Mamoru 岡山大学, 名誉教授 (70026772)
NARUSE Hiroshi 岡山大学, 教育学部, 教授 (20172596)
IKEDA Takeshi 岡山理科大学, 理学部, 准教授 (40309539)
NISHIMOTO Tetsu 近畿医療福祉大学, 社会福祉学部, 准教授 (80330520)
KAJI Shizuo 山口大学, 理学部, 講師 (00509656)
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| Project Period (FY) |
2009 – 2011
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| Keywords | Lie群 / 旗多様体 / コホモロジー / Chow環 / ループ空間 / アフィンGrassmann多様体 / Hopf代数 |
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| Research Abstract |
(1) We determined the ring structure of the integral cohomology ring of the flag manifold E_8/T, where E_8 denotes the compact simply-connected simple exceptional Lie group of rank 8 and T its maximal torus. We also identified the Schubert classes which generate this ring by means of the divided difference operators. Using this result, we were able to determine the Chow ring of the corresponding complex algebraic group E_8.
(2) Using the localization technique and the GKM description of the torus equivariant cohomology rings of homogeneous spaces, we computed the torus equivariant cohomology ring of the flag variety G_2/B and the complex quadric Q_n explicitly, where G_2 denotes the complex Lie group of type G_2 and B a Borel subgroup.
(3) Extending the result due to Kono-Kozima, we showed that the Pontrjagin ring of the based loop space of the infinite symplectic group Sp(resp. infinite orthogonal group SO) is isomorphic to the ring of Schur P-(resp. Q) functions
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