Algebraic Combinatorics of Symmetric Functions and its Applications to Representation Theory and Enumerative Combinatorics

• Project/Area Number: 18K03208
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Review Section: Basic Section 11010:Algebra-related
• 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)
• Co-Investigator(Kenkyū-buntansha): 石川 雅雄 岡山大学, 自然科学研究科, 教授 (40243373)
• Project Period (FY): 2018-04-01 – 2023-03-31
• Project Status: Completed (Fiscal Year 2022)
• Budget Amount: ¥4,420,000 (Direct Cost: ¥3,400,000、Indirect Cost: ¥1,020,000); Fiscal Year 2020: ¥1,430,000 (Direct Cost: ¥1,100,000、Indirect Cost: ¥330,000); Fiscal Year 2019: ¥1,430,000 (Direct Cost: ¥1,100,000、Indirect Cost: ¥330,000); Fiscal Year 2018: ¥1,560,000 (Direct Cost: ¥1,200,000、Indirect Cost: ¥360,000)
• Keywords: 対称関数 / 平面分割 / SchurのQ関数 / 中間斜交指標 / d-comoplete半順序集合 / rowmotion / 組合せ論 / 表現論 / 可積分系 / 半順序集合 / 凸多面体

Outline of Final Research Achievements

We study several aspects of symmetric function theory toward applications to combinatorics, representation theory and integrable systems. 1. We present several identities and positivety conjectures for Schur Q-functions associated to the root system of type C. 2. We find a Weyl-type character formula for intermediate symplectic characters and give an application to the enumeration of certain plane partitions. 3. We give affine Gordon-Bender-Knuth identities for cylindric Schur functions and apply them to the combinatorics of cylindric standard tableaux.

Academic Significance and Societal Importance of the Research Achievements

C型ルート系に付随したSchurのQ関数について，A型の場合と同様な関係式，正値性（予想）を見出した．これらの結果から背後に豊かな表現論的構造があることが期待され，新たな組合せ論，表現論を展開できる対象を提示できたとことに大きな意義がある．また，一連の研究においてパフィアンの絡んだ一般的な公式をいくつか与えることができ，幅広い分野に応用可能な手法を提供できた．

Research Products

• [Int'l Joint Research] Universitat Wien(オーストリア)
• [Int'l Joint Research] Sungkyunkwan University(韓国)
• [Int'l Joint Research] Sungkyunkwan University(韓国)
• [Int'l Joint Research] Universitat Wien(オーストリア)
• [Int'l Joint Research] University of Minnesota(米国)
• [Int'l Joint Research] University of Minnesota(米国)
• [Journal Article] Extended Schur’s Q-functions and the full Kostant-Toda hierarchy on the Lie algebra of type D (2023)
  • Author(s): Yuji Kodama and Soichi Okada
  • Journal Title: Physica D: Nonlinear Phenomena Volume: 443 Pages: 133589-133589
  • DOI: 10.1016/j.physd.2022.133589
  • Peer Reviewed / Int'l Joint Research
• [Journal Article] Minor summation formula and classical group characters of nearly rectangular shape (2022)
  • Author(s): 岡田 聡一
  • Journal Title: 京都大学数理解析研究所講究録 Volume: 2216
  • Open Access
• [Journal Article] Symplectic Q-functions (2021)
  • Author(s): Soichi Okada
  • Journal Title: J. Combin. Theory Ser. A Volume: 180 Pages: 105416-105416
  • DOI: 10.1016/j.jcta.2021.105416
  • Peer Reviewed
• [Journal Article] Intermediate symplectic characters and shifted plane partitions of shifted double staircase shape (2021)
  • Author(s): Soichi Okada
  • Journal Title: Comb. Theory Volume: 1 Issue: 0
  • DOI: 10.5070/c61055372
  • Peer Reviewed / Open Access
• [Journal Article] Intermediate symplectic characters and applications (2021)
  • Author(s): 岡田 聡一
  • Journal Title: 京都大学数理解析研究所講究録 Volume: 2183 Pages: 148-163
  • NAID: 40022685174
  • Open Access
• [Journal Article] Birational rowmotion and Coxeter-motion on minuscule posets (2021)
  • Author(s): S. Okada
  • Journal Title: Electron. J. Combin. Volume: 28 Issue: 1
  • DOI: 10.37236/9557
  • Peer Reviewed / Open Access
• [Journal Article] A generalization of Schur's P- and Q-functions (2020)
  • Author(s): S. Okada
  • Journal Title: Sem. Lothar. Combin. Volume: 81
  • Peer Reviewed / Open Access
• [Journal Article] Explicit formula for birational rowmotion on shifted staircases (2020)
  • Author(s): 岡田 聡一
  • Journal Title: 京都大学数理解析研究所講究録 Volume: 2161 Pages: 81-95
  • NAID: 40022336914
  • Open Access
• [Journal Article] A bialternant formula for odd symplectic characters and its application (2020)
  • Author(s): S. Okada
  • Journal Title: Josai Mathematical Monographs Volume: 12 Pages: 99-116
  • DOI: 10.20566/13447777_12_99
  • NAID: 120006839343
  • URL: https://libir.josai.ac.jp/il/meta_pub/G0000284repository_JOS-13447777-1209
  • Peer Reviewed / Open Access
• [Journal Article] Skew hook formula for d-complete posets via equivariant K-theory (2019)
  • Author(s): Naruse Hiroshi、Okada Soichi
  • Journal Title: Algebraic Combinatorics Volume: 2-4 Issue: 4 Pages: 541-571
  • DOI: 10.5802/alco.54
  • Peer Reviewed / Open Access
• [Journal Article] Pfaffian formulas and Schur Q-function identities (2019)
  • Author(s): S. Okada
  • Journal Title: Adv. Math. Volume: 353 Pages: 446-470
  • DOI: 10.1016/j.aim.2019.07.006
  • Peer Reviewed
• [Journal Article] Multivariate skew hook formula for d-complete posets (2019)
  • Author(s): 岡田 聡一
  • Journal Title: 京都大学数理解析研究所講究録 Volume: 2127 Pages: 79-93
  • NAID: 120006861838
  • Open Access
• [Journal Article] Birational Coxeter-motion on minuscule posets (2019)
  • Author(s): 岡田 聡一
  • Journal Title: 第 5 回 Algebraic Lie Theory and Representation Theory 報告集 Volume: - Pages: 219-233
• [Journal Article] Skew hook formula for d-complete posets via equivariant K-theory (extended abstract) (2019)
  • Author(s): H. Naruse and S. Okada
  • Journal Title: S\em. Lothar. Combin. Volume: 82B
  • Peer Reviewed / Open Access
• [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 Volume: 2075
  • Open Access
• [Presentation] Enumeration of shifted plane partitions of double staircase shape via intermediate symplectic characters (2022)
  • Author(s): Soichi Okada
  • Organizer: Joint Mathematics Meetings, AMS Special Session on Partition Theory and Related Topics
  • Int'l Joint Research / Invited
• [Presentation] Enumeration of standard tableaux and symmetric functions (2022)
  • Author(s): 岡田 聡一
  • Organizer: Japanese Conference on Combinatorics and its Applications 2022
  • Invited
• [Presentation] Positivity conjectures for symplectic Q-functions (2022)
  • Author(s): 岡田 聡一
  • Organizer: Toyama Workshop on Quantum Groups and Related Topics
  • Invited
• [Presentation] Symplectic Q-functions (2022)
  • Author(s): Soichi Okada
  • Organizer: Integrable Probability, Combinatorics and Representation Theory
  • Int'l Joint Research / Invited
• [Presentation] Affine Gordon-Bender-Knuth identities and related combinatorics (2022)
  • Author(s): 岡田 聡一
  • Organizer: 組合せ論的表現論における最近の展開
• [Presentation] Applications of the minor-summation formula to combinatorics and representation theory (2022)
  • Author(s): Soichi Okada
  • Organizer: Enumerative Combinatorics
  • Int'l Joint Research / Invited
• [Presentation] Enumeration of shifted plane partitions of shifted double staircase shape (2021)
  • Author(s): 岡田 聡一
  • Organizer: 離散数学とその応用研究集会 2021
• [Presentation] Minor summation formula and classical group characters of nearly rectangular shape (2021)
  • Author(s): 岡田 聡一
  • Organizer: 組合せ論的表現論および関連分野との連携
• [Presentation] Symplectic Q-functions (2020)
  • Author(s): S. Okada
  • Organizer: Algebraic Combinatorics Online Workshop
  • Int'l Joint Research / Invited
• [Presentation] A bialternant character formula for intermediate symplectic groups (2020)
  • Author(s): 岡田 聡一
  • Organizer: 離散数学とその応用研究集会 2020
• [Presentation] Schur's Q-functions and their generalizations (2020)
  • Author(s): 岡田 聡一
  • Organizer: 日本数学会2020年度秋季総合分科会（代数学分科会特別講演）
  • Invited
• [Presentation] Intermediate symplectic characters and applications (2020)
  • Author(s): 岡田 聡一
  • Organizer: 組合せ論的表現論の最近の進展
• [Presentation] Proof of birational file homomesy for minuscule posets (2020)
  • Author(s): S. Okada
  • Organizer: Dynamical Algebraic Combinatorics
  • Int'l Joint Research / Invited
• [Presentation] Positivity for symplectic Q-functions (2020)
  • Author(s): S. Okada
  • Organizer: AMS Special Session on Interactions between Combinatorics, Representation Theory, and Coding Theory, Joint Mathematics Meetings
  • Int'l Joint Research / Invited
• [Presentation] Positivity for symplectic Q-functions (2019)
  • Author(s): S. Okada
  • Organizer: 82th Seminaire Lotharingien de Combinatoire
  • Int'l Joint Research
• [Presentation] Generalization of Schur's Q-functions (2019)
  • Author(s): S. Okada
  • Organizer: New Interactions between Geometry and Combinatorics
  • Int'l Joint Research / Invited
• [Presentation] Birational Coxeter-motion on minuscule posets (2019)
  • Author(s): 岡田 聡一
  • Organizer: 第 5 回 Algebraic Lie Theory and Representation Theory
• [Presentation] Skew hook formula for d-complete posets via equivariant K-theory (2019)
  • Author(s): S. Okada
  • Organizer: 31st International Conference on Formal Power Series and Algebraic Combinatorics, Poster Session
  • Int'l Joint Research
• [Presentation] Birational rowmotion on minuscule posets (2019)
  • Author(s): 岡田 聡一
  • Organizer: 組合せ論サマースクール 2019
• [Presentation] ミニスキュール半順序集合上の双有理版 rowmotion と双有理版 Coxetermotion (2019)
  • Author(s): 岡田 聡一
  • Organizer: 日本数学会 2019 年度秋季総合分科会無限可積分系特別セッション
• [Presentation] Explicit formula for birational rowmotion on shifted staircases (2019)
  • Author(s): 岡田 聡一
  • Organizer: 表現論とその組合せ論的側面
• [Presentation] Birational rowmotion and Coxeter-motion on minuscule posets (2019)
  • Author(s): S. Okada
  • Organizer: Workshop on Algebraic and Enumerative Combinatorics
  • Int'l Joint Research / Invited
• [Presentation] Symplectic Q-functions (2019)
  • Author(s): S. Okada
  • Organizer: Taipei International Workshop on Combinatorics and Graph Theory
  • Int'l Joint Research / Invited
• [Presentation] Birational rowmotion and Coxeter-motion on minuscule posets (2019)
  • Author(s): S. Okada
  • Organizer: AMS Spring Central and Western Joint Sectional Meeting, Special Session on Dynamical Systems and Algebraic Combinatorics
  • Int'l Joint Research / Invited
• [Presentation] Skew hook formula for d-complete posets (2018)
  • Author(s): S. Okada
  • Organizer: Japanese Conference on Combinatorics and its Applications
  • Int'l Joint Research
• [Presentation] Generalized parking spaces (2018)
  • Author(s): S. Okada
  • Organizer: Representation Theory Days 2018
  • Int'l Joint Research / Invited
• [Presentation] Multivariate skew hook formula for d-complete posets (2018)
  • Author(s): 岡田 聡一
  • Organizer: 組合せ論的表現論の諸相