Macdonald polynomials, multivariable hypergeometric series and their application to lattice models

• Project/Area Number: 20540203
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Single-year Grants
• Section: 一般
• Research Field: Global analysis
• Research Institution: The University of Tokyo
• Principal Investigator: SHIRAISHI Junichi 東京大学, 大学院・数理科学研究科, 准教授 (20272536)
• Project Period (FY): 2008 – 2011
• Project Status: Completed (Fiscal Year 2011)
• Budget Amount: ¥3,380,000 (Direct Cost: ¥2,600,000、Indirect Cost: ¥780,000); Fiscal Year 2011: ¥780,000 (Direct Cost: ¥600,000、Indirect Cost: ¥180,000); Fiscal Year 2010: ¥780,000 (Direct Cost: ¥600,000、Indirect Cost: ¥180,000); Fiscal Year 2009: ¥780,000 (Direct Cost: ¥600,000、Indirect Cost: ¥180,000); Fiscal Year 2008: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000)
• Keywords: 可能積分系 / マクドナルド多項式 / 超幾何級数 / Ding-Iohara代数 / topological vertex / Nekrasov分配関数 / 可積分系 / Din-Iohara代数 / 変形W代数 / Koornwinder多項式 / 可解格子模型 / 楕円量子群 / Macdonald多項式 / フュージョン則

Research Abstract

I studied the representation theory of the Ding-Iohara algebra, clarifying a deep connection with the Macdonald polynomials. It was found that certain matrix elements of the homomorphisms of Ding-Iohara (or deformed Virasoro/W) algebra can be written as multivariable hypergeometric series, and are eigenfunctions of the Macdoanld difference operator. The series have a duality between the coordinate variables and momentum ones. For the Macdonald polynomials of type D and C, with one row partitions, similar hypergeometric series expressions are obtained by using the deformed W algebras.

Research Products

• [Journal Article] Quantum algebraic approach to refined topological vertex (2012)
  • Author(s): H. Awata, B. Feigin and J. Shiraishi
  • Journal Title: J. High Energy Phys Volume: no. 3 Issue: 3 Pages: 41-41
  • DOI: 10.1007/jhep03(2012)041
  • Peer Reviewed
• [Journal Article] On some special solutions to periodic Binjamin-Ono equation with discreteLaplacian (2012)
  • Author(s): Y. Tutiya and J. Shiraishi
  • Journal Title: Math. Comput. Simulation 82 Volume: no. 7 Pages: 1341-1347
• [Journal Article] Periodic Benjamin-Ono equation with discrete Laplacian and 2D-Toda hierarchy.Newtrends in quantum integrable systems (2011)
  • Author(s): J.Shiraishi and Y. Tutiya
  • Journal Title: World Sci.Publ., Hackensack, NJ, Pages: 357-371
• [Journal Article] Notes on Ding-Iohara algebra and AGT conjecture (2011)
  • Author(s): H. Awata, B. Feigin, A. Hoshino, M. Kanai. J. Shiraishi and S. Yanagida
  • Journal Title: 京都大学数理解析研究所講究録 Volume: 1765 Pages: 12-32
• [Journal Article] Notes on Ding-Iohara algebra andAGT conjecture (2011)
  • Author(s): H.Awata, B.Feigin, A.Hoshino, M.Kana, J.Shiraishi, S.Yanagida
  • Journal Title: RIMS講究録 Volume: 1765 Pages: 12-32
• [Journal Article] Kernel function and quantum algebras (2010)
  • Author(s): B. Feigin, A. Hoshino, J. Shibahara, J. Shiraishi and S. Yanagida
  • Journal Title: 京都大学数理解析研究所講究録 Volume: 1689 Pages: 133-152
• [Journal Article] Periodic Benjamin-Ono equation with discrete Laplacian and 2D-Toda Hierarchy,In New Trenda inQuantum Integrable Systems (2010)
  • Author(s): J.Shiraishi, Y.Tutiya
  • Journal Title: Proceeding of the Infinite Analysis 09(Editors B.Feigin, M.Jimbo and M.Okado)(World Scientific) Pages: 357-371
• [Journal Article] Kernel function and quantum algebras (2010)
  • Author(s): B.Feigin, A.Hoshino, J.Shibahara, J.Shiraishi, S.Yanagida
  • Journal Title: 数理科学研究所講究録 Volume: 1689 Pages: 133-152
• [Journal Article] Periodic Benjamin-Ono equation with discrete Laplacian and 2D-Zbda Hierarchy (2010)
  • Author(s): J.Shiraishi Y.Tutiya
  • Journal Title: New Trends in Quantum Integrable Systems, Proceeding of the Infinite Analysis 09, editors B.Feigin, M.Jimbo M.Okado Pages: 357-371
• [Journal Article] The integrals of motion for the elliptic defoemation of the Virasoro algebraand WN algebra (2009)
  • Author(s): T.Kojima and J.Shiraishi
  • Journal Title: Nonlinear Anal. 71 Volume: no. 12
• [Journal Article] Periodic ILW equation with discrete Laplacian (2009)
  • Author(s): J. Shiraishi and Y. Yutiya
  • Journal Title: J. Phys. A Volume: 42no. 40 Pages: 404018-15
• [Journal Article] A commutative algebra on degenerate CP1 and Macdonald polynomials (2009)
  • Author(s): B. Feigin, K. Hashizume, A. Hoshino, J. Shiraishi and S. Yanagida
  • Journal Title: J. Math. Phys. 50 Volume: no. 9 Pages: 095215-42
• [Journal Article] Remark on the integrals of motion associated with level k realization of U_qp(sl2^) (2009)
  • Author(s): T. Kojima and J. Shiraishi
  • Journal Title: J. Geom. Symmetry Phys Volume: 14 Pages: 35-49
• [Journal Article] Kernel functions for difference operators of Ruijsenaars type and their applications (2009)
  • Author(s): Y. Komori, M. Noumi and J. Shiraishi
  • Journal Title: SIGMA Symmetry Integrability Geom. Methods Appl. 5 Volume: 054 Pages: 40-40
  • DOI: 10.3842/sigma.2009.054
• [Journal Article] Periodic ILW equation with discrete Laplacian (2009)
  • Author(s): J.Shiraishi, Y.Tutiya
  • Journal Title: J.Phys.A 42 no.40
• [Journal Article] A commutative algebra on degenerate CP^1 and Macdonald polynomials (2009)
  • Author(s): B.Feigin, K.Hashizume, A.Hoshino, J.Shiraishi, S.Yanagida
  • Journal Title: J.Math.Phys.50 no.9
• [Journal Article] Kernel functions for difference operators of Ruijsenaars type and their applications. (2009)
  • Author(s): Y.Komori, M.Noumi, J.Shiraishi
  • Journal Title: SIGMA Symmetry Integrability Geom.Methods Appl.5 054 Pages: 40-40
• [Journal Article] A commutative algebra on degenerate CP and Macdonald polynomials (2009)
  • Author(s): B.Feigin, K.Hashizume, A.Hoshino, J.Shiraishi, S.Yanagida
  • Journal Title: SIGMA Symmetry Integrability Geom.Methods Appl. 5 Pages: 095215-42
  • Peer Reviewed
• [Journal Article] Periodic ILW equation with discrete Laplacian (2009)
  • Author(s): J.Shiraishi, Y.Tutiya
  • Journal Title: J.Phys.A 42 Pages: 404018-15
  • Peer Reviewed
• [Journal Article] The Integrals of motion for the deformed Walgebra W_qt(glN^). II. Proof of the commutation relations (2008)
  • Author(s): T.Kojima and J.Shiraishi
  • Journal Title: Comm. Math.Phys. 283 Volume: no. 3 Pages: 795-851
• [Presentation] Ding-Iohara代数のFock moduleの族と3点intertwining operator (2012)
  • Author(s): 粟田, Feigin, 白石
  • Organizer: 2012年日本数学会年会
  • Place of Presentation: 東京理科大学
  • Year and Date: 2012-03-29
• [Presentation] Ding-Iohara代数のn点関数とNekrasov分配関数 (2012)
  • Author(s): 粟田, Feigin, 白石
  • Organizer: 2012年日本数学会年会
  • Place of Presentation: 東京理科大学
  • Year and Date: 2012-03-29
• [Presentation] Ding-Iohara代数のprimary場の因子化公式I (2011)
  • Author(s): 粟田, Feigin, 星野, 金井, 白石, 柳田
  • Organizer: 2011年日本数学会秋期総合分科会
  • Place of Presentation: 信州大学
  • Year and Date: 2011-10-01
• [Presentation] Ding-Iohara代数のprimary場の因子化公式II-Pieri係数,Nekrasov因子と梶原のEuler変換 (2011)
  • Author(s): 粟田, Feigin, 星野, 金井, 白石, 柳田
  • Organizer: 2011年日本数学会秋期総合分科会
  • Place of Presentation: 信州大学
  • Year and Date: 2011-10-01
• [Presentation] Ding-Iohara代数のprimary場の因子化公式III (2011)
  • Author(s): 粟田, Feigin, 星野, 金井, 白石, 柳田
  • Organizer: 2011年日本数学会秋期総合分科会
  • Place of Presentation: 信州大学
  • Year and Date: 2011-10-01
• [Presentation] Macdonald polynomials and quantum algebras (2010)
  • Author(s): J.Shiraishi
  • Organizer: 15th Itzykson meeting, Nwe trends in quantum integrability
  • Place of Presentation: IPhT Saclay,France
  • Year and Date: 2010-06-23
• [Presentation] 15th Itzykson meeting, Nwe trends in quantum integrability (2010)
  • Author(s): Macdonald polynomials and quantum algebras
  • Organizer: IPhT Saclay
  • Place of Presentation: France
  • Year and Date: 2010-06-23
• [Presentation] Macdonald polynomials and quantum algebras (2010)
  • Author(s): J.Shiraishi
  • Organizer: 15th Itzykson meeting, Nwe trends in quantum integrability
  • Place of Presentation: IPhT Saclay, France
  • Year and Date: 2010-06-23
• [Presentation] Macdonald polynomials and quantum algebras (2009)
  • Author(s): 白石潤一
  • Organizer: RIMS研究集会 表現論と組合わせ論
  • Place of Presentation: 北海道大学理学部
  • Year and Date: 2009-08-27
• [Presentation] Infinite Analysis 09, New Trends in Quantum Integrable Systems (2009)
  • Author(s): Hirota-Miwa equations, Macdonald operators
  • Organizer: Department of Mathematics Kyoto University
  • Place of Presentation: Japan
  • Year and Date: 2009-07-30
• [Presentation] Macdonald多項式と可積分系 (2008)
  • Organizer: 2008年日本数学会秋期総合分科会特別講演
  • Place of Presentation: 東京工業大学
  • Year and Date: 2008-09-25
• [Presentation] 日本数学会秋期総合分科会 (2008)
  • Author(s): Macdonald多項式と可積分系
  • Place of Presentation: 東京工業大学(特別講演)
  • Year and Date: 2008-09-25
• [Presentation] Workshop "Integrable quantum systems and solvable statistical mechanical models (2008)
  • Author(s): Macdonald polynomials and integrals of motion
  • Organizer: CRM Centre de Recherches Math\'ematiques, Montreal
  • Place of Presentation: Canada
  • Year and Date: 2008-07-04
• [Presentation] Ding-Iohara代数のprimary場の因子化公式I
  • Author(s): 粟田英資、B.Feigin,星野歩、金井政宏、白石潤一、柳田伸太郎
  • Organizer: 2011年度日本数学会秋季総合分科会
  • Place of Presentation: 信州大学
• [Presentation] Ding-Iohara代数のprimary場の因子化公式II-Pieri係数、Nekrasov因子と梶原のEuler変換
  • Author(s): 粟田英資、B.Feigin,星野歩、金井政宏、白石潤一、柳田伸太郎
  • Organizer: 2011年度日本数学会秋季総合分科会
  • Place of Presentation: 信州大学
• [Presentation] Ding-Iohara代数のprimary場の因子化公式III
  • Author(s): 粟田英資、B.Feigin,星野歩、金井政宏、白石潤一、柳田伸太郎
  • Organizer: 2011年度日本数学会秋季総合分科会
  • Place of Presentation: 信州大学
• [Presentation] 周期境界Benjamin-Ono方程式のある拡張について
  • Author(s): 土谷洋平、白石潤一
  • Organizer: 2010年度日本数学会秋季総合分科会
  • Place of Presentation: 名古屋大学
• [Presentation] 周期境界intermediateling wave方程式のある拡張について
  • Author(s): 土谷洋平、白石潤一
  • Organizer: 2010年度日本数学会秋季総合分科会
  • Place of Presentation: 名古屋大学
• [Presentation] Ding-Iohara代数のレベル0表現とMacdonald差分作用素について
  • Author(s): B.Feigin,橋爪清史、星野歩、白石潤一、柳田伸太郎
  • Organizer: 2010年度日本数学会秋季総合分科会
  • Place of Presentation: 名古屋大学
• [Presentation] Ding-Iohara代数のレベルm表現と変形W(sl_)代数について
  • Author(s): B.Feigin,橋爪清史、星野歩、白石潤一、柳田伸太郎
  • Organizer: 2010年度日本数学会秋季総合分科会
  • Place of Presentation: 名古屋大学