| Project/Area Number |
24540010
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Multi-year Fund |
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| Section | 一般 |
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| Research Field |
Algebra
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| Research Institution | Tokyo Institute of Technology |
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Principal Investigator |
Naito Satoshi 東京工業大学, 理工学研究科, 教授 (60252160)
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| Co-Investigator(Renkei-kenkyūsha) |
SAITO Yoshihisa 東京大学, 大学院数理科学研究科, 准教授 (20294522)
KATO Syu 京都大学, 大学院理学研究科, 准教授 (40456760)
SAGAKI Daisuke 筑波大学, 数理物質系, 准教授 (40344866)
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| Research Collaborator |
Lenart Cristian State University of New York at Albany, Department of Mathematics and Statistics, 教授
Schilling Anne University of California, Department of Mathematics, 教授
Shimozono Mark Virginia Tech, Department of Mathematics, 教授
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| Project Period (FY) |
2012-04-01 – 2016-03-31
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| Project Status |
Completed (Fiscal Year 2016)
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| Budget Amount *help |
¥4,940,000 (Direct Cost: ¥3,800,000、Indirect Cost: ¥1,140,000)
Fiscal Year 2016: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000)
Fiscal Year 2015: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000)
Fiscal Year 2014: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000)
Fiscal Year 2013: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000)
Fiscal Year 2012: ¥780,000 (Direct Cost: ¥600,000、Indirect Cost: ¥180,000)
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| Keywords | 表現論 / アフィン量子群の表現論 / レベル・ゼロ表現 / extremal ウエイト加群 / Demazure 部分加群 / マクドナルド多項式 / Lakshmibai-Seshadri パス / 半無限旗多様体 / 代数学 / Macdonald 多項式 / pQLS パス / Ｍａｃｄｏｎａｌｄ ｐｏｌｙｎｏｍｉａｌ / ｃｒｙｓｔａｌ ｂａｓｉｓ / ｑｕａｎｔｕｍ ａｆｆｉｎｅ ａｌｇｅｂｒａ / ｅｘｔｒｅｍａｌ ｗｅｉｇｈｔ ｍｏｄｕｌｅ / Ｌａｋｓｈｍｉｂａｉ－Ｓｅｓｈａｄｒｉ ｐａｔｈ / ｑｕａｎｔｕｍ Ｗｅｙｌ ｍｏｄｕｌｅ / 結晶基底 / semi-infinite Bruhat 順序 / 量子 Bruhat グラフ / Demazure 加群 |
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| Outline of Final Research Achievements |
First, we got an explicit description, in terms of the quantum Bruhat graph, of the graded character of an arbitrary Demazure submodule of a level-zero extremal weight module over a quantum affine algebra. Also, we got an explicit description, in terms of the quantum Bruhat graph, of the specializations at t = 0 and t = infinity of an arbitrary nonsymmetric Macdonald polynomial. By combining these results, we proved that the graded character of the Demazure submodule corresponding to the identity element (resp., the longest element) of a finite Weyl group is identical to the product of a certain factor (which is an explicit rational function in q) and the specialization at t = 0 (resp., at t = infinity) of the symmetric (resp., nonsymmetric) Macdonald polynomial associated to a dominant integral weight (resp., anti-dominant integral weight).
Moreover, we studied the connection of level-zero Demazure submodules above with Schubert subvarieties of a semi-infinite flag manifold.
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