Construction of affine geometric crystals and Representation theory of crystal bases

2009 Fiscal Year Self-evaluation Report

• Project/Area Number: 19540050
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Single-year Grants
• Section: 一般
• Research Field: Algebra
• Research Institution: Sophia University
• Principal Investigator: NAKASHIMA Toshiki Sophia University, 理工学部, 教授 (60243193)
• Project Period (FY): 2007 – 2009
• Keywords: 結晶基底 / 幾何結晶 / 熱帯化 / 超離散化 / ヘッケ環 / 代数群 / Tropical R map / 保型形式

Research Abstract

(1)Affine Kac-Moody群のunipotnent部分群上もしくは、可積分加群上のある種の不変関数を求めることから始める。群上の不変関数については群論の専門家である筱田との協力により研究を遂行する。
(2)ソリトン理論における無限次元多様体の関係から幾何結晶との結びつきについての考察を行ってみたい。
(3)可能であれば初年度中にSteinberg多様体上の幾何結晶の構成にも着手する。
Steinberg多様体はHecke環の表現論と密接に結びついており、Hecke環の表現論の専門家である五味と共同で幾何結晶の構成に取り組みたい。

Research Products

• [Journal Article] Affine Geometric Crystal of type D^{(3)}_4. (2010)
  • Author(s): 中島俊樹(M. Igarashi)
  • Journal Title: Contemporary Mathematics 506 Pages: 215-226
  • Peer Reviewed
• [Journal Article] Universal tropical R map of sl(2) and prehomogeneous geometric crystals. (2009)
  • Author(s): 中島俊樹
  • Journal Title: RIMS Kokyuroku Bessatsu B11 Pages: 101-116
  • Peer Reviewed
• [Journal Article] Affine Geometric Crystals and Limit of Perfect Crystals. (2008)
  • Author(s): 中島俊樹, (M. Kashiwara, M. Okado)
  • Journal Title: Transactions in American Mathematical Society 30 Pages: 3645-3687
  • Peer Reviewed
• [Journal Article] Affine Geometric Crystal of type G^{(1)}_2. (2007)
  • Author(s): 中島俊樹
  • Journal Title: Contemporary Mathematics 442 Pages: 179-192
  • Peer Reviewed
• [Presentation] Admissible Pictures and Littlewood-Richardson Crystals. (2009)
  • Author(s): 中島俊樹
  • Organizer: International Workshop on Combinatorial and Geometric approach to Representation Theory
  • Place of Presentation: Seoul National University, Korea
  • Year and Date: 2009-09-21
• [Presentation] Epsilon Systems of geometric Crystals. (2009)
  • Author(s): 中島俊樹
  • Organizer: Geometric Aspects of Discrete and Ultra-Discrete Integrable Systems, (A satellite meeting of the Isaac Newton Institute Programme on Desicrete Integrable Systems)
  • Place of Presentation: The University of Glasgow, United Kingdom.
  • Year and Date: 2009-03-31
• [Presentation] Geometric Crystals on flag varieties andmaximal unipotent subgroups. (2008)
  • Author(s): 中島俊樹
  • Organizer: Workshop, 'Crystals and Tropical Combinatorics'
  • Place of Presentation: Kansai Seminar House, 京都
  • Year and Date: 2008-08-30
• [Presentation] Affine Tropical R and Prehomogeneous Geometric Crystals. (2008)
  • Author(s): 中島俊樹
  • Organizer: Quantum Affine Lie Algebras, Extended Affine Lie Algebras, and Applications
  • Place of Presentation: Banff International Research Stations
  • Year and Date: 2008-03-05
• [Presentation] Uniqueness of tropical R' (2007)
  • Author(s): 中島俊樹
  • Organizer: 組合せ論的表現論の拡がり
  • Place of Presentation: 京都大学数理解析研究所
  • Year and Date: 2007-10-25