Research on combinatorics with representation theory related to leaf posets and surrounding topics

• Project/Area Number: 23540017
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Algebra
• Research Institution: Wakayama University
• Principal Investigator: TAGAWA Hiroyuki 和歌山大学, 教育学部, 教授 (80283943)
• Project Period (FY): 2011-04-28 – 2016-03-31
• Project Status: Completed (Fiscal Year 2015)
• Budget Amount: ¥5,070,000 (Direct Cost: ¥3,900,000、Indirect Cost: ¥1,170,000); Fiscal Year 2015: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000); Fiscal Year 2014: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000); Fiscal Year 2013: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000); Fiscal Year 2012: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000); Fiscal Year 2011: ¥1,430,000 (Direct Cost: ¥1,100,000、Indirect Cost: ¥330,000)
• Keywords: hook length poset / d-complete poset / leaf poset / 行列式 / 超幾何級数 / Askey-Wilson 多項式 / Aztec rectangle

Outline of Final Research Achievements

In this research, we mainly obtained the following results: We found that the d-complete poset being multivariable hook length poset implies the leaf poset being multivariable hook length poset. We expressed the number of the domino tilings in the Aztec rectangle with connected 2r-holes by a determinant of the matrix of size 2r whose elements are hypergeometric series. We proved a quadratic formula of hypergeometirc series which is useful for the proof of some kind of Hankel type determinants.

Research Products

• [Journal Article] Domino tilings of Aztec rectangles with connected holes (2016)
  • Author(s): Masao Ishikawa, Fumihiko Nakano, Taizo Sadahiro, Hiroyuki Tagawa
  • Journal Title: 数理解析研究所講究録 Volume: 印刷中
  • Open Access / Acknowledgement Compliant
• [Journal Article] A quadratic formula for basic hypergeometric series related to Askey-Wilson polynomials (2015)
  • Author(s): Victor J.W.Guo, Masao Ishikawa, Hiroyuki Tagawa, Jiang Zeng
  • Journal Title: Proceedings of the American Mathematical Society Volume: 143 Issue: 5 Pages: 2003-2015
  • DOI: 10.1090/s0002-9939-2015-12099-0
  • Peer Reviewed / Int'l Joint Research / Acknowledgement Compliant
• [Journal Article] An extension of Wilson's Gram determinants (2015)
  • Author(s): 石川雅雄・田川裕之
  • Journal Title: 数理解析研究所講究録 Volume: 1945 Pages: 38-53
  • Open Access / Acknowledgement Compliant
• [Journal Article] Leaf posets and multivariate hook length property (2014)
  • Author(s): 石川雅雄・田川裕之
  • Journal Title: 数理解析研究所講究録 Volume: 1913 Pages: 67-80
  • Acknowledgement Compliant
• [Journal Article] A generalization of the Mehta-Wang determinant and Askey-Wilson polynomials (2013)
  • Author(s): Victor J. W. Guo, Masao Ishikawa, Hiroyuki Tagawa, Jiang Zeng
  • Journal Title: 25th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2013), DMTCS proc. AS Pages: 749-760
  • Peer Reviewed
• [Journal Article] A generalization of the Mehta-Wang determinant and the Askey-Wilson polynomials (2012)
  • Author(s): 石川雅雄・田川裕之
  • Journal Title: 数理解析研究所講究録 Volume: 1795 Pages: 204-223
• [Presentation] d-complete poset と leaf poset (2016)
  • Author(s): 田川裕之
  • Organizer: 2015年度表現論ワークショップ
  • Place of Presentation: 県民ふれあい会館（鳥取県鳥取市）
  • Year and Date: 2016-01-07
• [Presentation] A generalization of the Mehta-Wang determinant and the Askey-Wilson polynomials (2011)
  • Author(s): 石川雅雄・田川 裕之
  • Organizer: 研究集会「組み合わせ論的表現論の拡がり」
  • Place of Presentation: 京都大学数理解析研究所
• [Presentation] A generalization of the Mehta-Wang determinant and Askey-Wilson polynomials
  • Author(s): Victor J. W. Guo, Masao Ishikawa, Hiroyuki Tagawa, Jiang Zeng
  • Organizer: 25th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2013)
  • Place of Presentation: Paris, France
• [Presentation] An extension of Wilson’s Gram determinants
  • Author(s): 石川雅雄, 田川裕之
  • Organizer: RIMS研究集会「組合せ論的表現論の展望」
  • Place of Presentation: 京都大学数理解析研究所
• [Presentation] Leaf posets and multivariate hook length property
  • Author(s): 石川雅雄・田川裕之
  • Organizer: RIMS合宿型セミナー「ヤング図形・統計物理に関連する代数的組合せ論」
  • Place of Presentation: 国際高等研究所
• [Remarks] The home page of Hiroyuki TAGAWA
  • URL: http://www.center.wakayama-u.ac.jp/~tagawa/
• [Remarks] The home page of Hiroyuki TAGAWA
  • URL: http://www.center.wakayama-u.ac.jp/~tagawa
• [Remarks] The home page of Hiroyuki TAGAWA
  • URL: http://www.center.wakayama-u.ac.jp/~tagawa/
• [Remarks]
  • URL: http://www.center.wakayama-u.ac.jp/~tagawa/