Representations of double affine Hecke algebras and their applications to integrable systems

• Project/Area Number: 21740005
• Research Category: Grant-in-Aid for Young Scientists (B)
• Allocation Type: Single-year Grants
• Research Field: Algebra
• Research Institution: The University of Tokyo
• Principal Investigator: KASATANI Masahiro The University of Tokyo, 大学院・数理科学研究科, 特任助教 (40527884)
• Project Period (FY): 2009 – 2010
• Project Status: Completed (Fiscal Year 2010)
• Budget Amount: ¥1,950,000 (Direct Cost: ¥1,500,000、Indirect Cost: ¥450,000); Fiscal Year 2010: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000); Fiscal Year 2009: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000)
• Keywords: アフィンヘッケ環 / qKZ方程式 / Macdonald多項式 / Koornwinder多項式

Research Abstract

(1) We formulated boundary qKZ equation in terms of the polynomial representation of the double affine Hecke algebra (DAHA) of type C^∨C. We gave a procedure to construct its polynomial solutions from solutions for an eigenvalue problem. Some special solutions are constructed by non-symmetric Koornwinder polynomials. (2) We confirmed that it will be possible to formulate a construction of polynomial solutions for a degenerate limit of the qKZ equation (with no boundary) in terms of the polynomial representation of degenerate DAHA. (3) We suggest an equality between non-symmetric Macdonald polynomials at roots of unity and Schur symmetric polynomials.

Research Products

• [Journal Article] Boundary Quantum Knizhnik-Zamolodchikov Equation (2010)
  • Author(s): Masahiro Kasatani
  • Journal Title: New Trends in Quantum Integrable Systems : Proceedings of the Infinite Analysis 09 Pages: 157-171
  • Peer Reviewed
• [Presentation] C∨C型DAHAの多項式表現と境界付きqKZ方程式について (2010)
  • Author(s): 笠谷昌弘
  • Organizer: 研究集会「BC系とAGT予想の周辺」
  • Place of Presentation: 東京大学大学院数理科学研究科
  • Year and Date: 2010-09-13
• [Presentation] $C^/vee C$型DAHAの多項式表現と境界付きqKZ方程式について (2010)
  • Author(s): 笠谷昌弘
  • Organizer: 研究集会「BC系とAGT予想の周辺」
  • Place of Presentation: 東京大学大学院 数理科学研究科
  • Year and Date: 2010-09-13
• [Presentation] 境界付きqKZ方程式と非対称Koornwinder多項式について (2010)
  • Author(s): 笠谷昌弘
  • Organizer: RIMS研究集会2010「可積分系数理の多様性」
  • Place of Presentation: 京都大学数理解析研究所
  • Year and Date: 2010-08-19
• [Presentation] The qKZ equation and its polynomial solutions (2010)
  • Author(s): Masahiro Kasatani
  • Organizer: RIMS合宿型セミナー「Diagram algebras and related topics」
  • Place of Presentation: 沖縄県宜野湾市
  • Year and Date: 2010-07-09
• [Presentation] アフィンヘッケ環の多項式表現とqKZ方程式について (2009)
  • Author(s): 笠谷昌弘
  • Organizer: 日本数学会2009年度秋季総合分科会 無限可積分系セッション特別講演
  • Place of Presentation: 大阪大学
  • Year and Date: 2009-09-24
• [Presentation] アフィンヘッケ環の多項式表現とqKZ方程式について (2009)
  • Author(s): 笠谷昌弘
  • Organizer: 日本数学会2009年度秋季総合分科会 無限可積分系セソション特別講演
  • Place of Presentation: 大阪大学
  • Year and Date: 2009-09-24
• [Presentation] Boundary quantum Knizhnik-Zamolodchikov equation (2009)
  • Author(s): Masahiro Kasatani
  • Organizer: Infinite Analysis 09 -New Trends in Quantum Integrable Systems-
  • Place of Presentation: 京都大学
  • Year and Date: 2009-07-30