Combinatorics of special polynomials associated to certain solutions of Painleve equations

• Project/Area Number: 15K13425
• Research Category: Grant-in-Aid for Challenging Exploratory Research
• Allocation Type: Multi-year Fund
• Research Field: Algebra
• Research Institution: Nagoya University
• Principal Investigator: Okada Soichi 名古屋大学, 多元数理科学研究科, 教授 (20224016)
• Project Period (FY): 2015-04-01 – 2018-03-31
• Project Status: Completed (Fiscal Year 2017)
• Budget Amount: ¥3,900,000 (Direct Cost: ¥3,000,000、Indirect Cost: ¥900,000); Fiscal Year 2017: ¥1,300,000 (Direct Cost: ¥1,000,000、Indirect Cost: ¥300,000); Fiscal Year 2016: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000); Fiscal Year 2015: ¥1,430,000 (Direct Cost: ¥1,100,000、Indirect Cost: ¥330,000)
• Keywords: Painleve 方程式 / KP 階層 / 対称関数 / Schur 関数 / Schur Q 関数 / 組合せ論 / Painleve方程式

Outline of Final Research Achievements

We study several aspects of integrable systems and symmetric functions, toward unraveling combinatorial structure of special polynomials associated to certain solutions of the Painleve-type equations. 1. We find Giambelli-type determinant identities, which characterize the expansion coefficients of the \tau-function of the KP hierarchy, and give new determinant expressions for Schur functions. 2. We find a relation between two generalizations of Schur Q-functions corresponding to rational Schur functions. Also we give several identities and positivity conjectures for Schur Q-functions associated to the root system of type C.

Research Products

• [Journal Article] Pieri rules for symplectic and factorial Q-functions (2018)
  • Author(s): 岡田 聡一
  • Journal Title: 数理解析研究所講究録「組合せ論と表現論」 Volume: 印刷中
  • Open Access
• [Journal Article] On the expansion coefficients of KP tau function (2017)
  • Author(s): Atsushi Nakayashiki, Soichi Okada, Yoko Shigyo
  • Journal Title: Journal of Integrable Systems Volume: 2 Issue: 1 Pages: 1-23
  • DOI: 10.1093/integr/xyx007
  • Peer Reviewed / Open Access
• [Journal Article] Symplectic Q-functions (2017)
  • Author(s): 岡田 聡一
  • Journal Title: 数理解析研究所講究録「リー型の組合せ論」 Volume: 印刷中
  • Open Access
• [Journal Article] Schur Q-functions and symplectic Q-functions (2016)
  • Author(s): 岡田 聡一
  • Journal Title: 2016 年度表現論シンポジウム講演集 Volume: - Pages: 111-132
  • NAID: 130008005018
  • Open Access
• [Presentation] d-Complete posets and hook formulas (2018)
  • Author(s): Soichi Okada
  • Organizer: Algebraic and Enumerative Combinatorics in Okayama
  • Int'l Joint Research / Invited
• [Presentation] Skew hook formula for d-complete posets (2018)
  • Author(s): Soichi Okada
  • Organizer: 80th Seminaire Lotharingien de Combinatoire
  • Int'l Joint Research
• [Presentation] KP hierarchy and Giambelli identities (2017)
  • Author(s): 岡田 聡一
  • Organizer: 可積分系の数理と応用
• [Presentation] Pieri rules for factorial and symplectic Schur Q-functions (2017)
  • Author(s): 岡田 聡一
  • Organizer: 表現論と組合せ論
• [Presentation] Symplectic Schur Q-functions (2017)
  • Author(s): Soichi Okada
  • Organizer: Workshop on Enumerative Combinatorics
  • Int'l Joint Research / Invited
• [Presentation] Schur functions and Schur Q-functions (2016)
  • Author(s): 岡田 聡一
  • Organizer: 2016 年度表現論シンポジウム
  • Place of Presentation: 沖縄
  • Year and Date: 2016-12-01
  • Invited
• [Presentation] Symplectic Q-functions (2016)
  • Author(s): 岡田 聡一
  • Organizer: Lie 型の組合せ論
  • Place of Presentation: 京都
  • Year and Date: 2016-10-04
  • Invited
• [Presentation] Pfaffian identities and Q-functions (2016)
  • Author(s): Soichi Okada
  • Organizer: 76th Seminaire Lotharingien de Combinatoire
  • Place of Presentation: Ottrott, France
  • Year and Date: 2016-04-04
  • Int'l Joint Research
• [Presentation] Pfaffian identities and Schur's Q-functions (2016)
  • Author(s): Soichi Okada
  • Organizer: Statistical Mechanics and Combinatorics
  • Place of Presentation: Simons Center for Geometry and Physics, Stony Brook, U.S.A.
  • Year and Date: 2016-03-09
  • Invited