Study on random partitions and random matrices based on combinatorics and representation theory

• Project/Area Number: 17K05281
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Basic analysis
• Research Institution: Kagoshima University
• Principal Investigator: Matsumoto Sho 鹿児島大学, 理工学域理学系, 准教授 (60547553)
• Project Period (FY): 2017-04-01 – 2023-03-31
• Project Status: Completed (Fiscal Year 2022)
• Budget Amount: ¥3,900,000 (Direct Cost: ¥3,000,000、Indirect Cost: ¥900,000); Fiscal Year 2021: ¥780,000 (Direct Cost: ¥600,000、Indirect Cost: ¥180,000); Fiscal Year 2020: ¥780,000 (Direct Cost: ¥600,000、Indirect Cost: ¥180,000); Fiscal Year 2019: ¥780,000 (Direct Cost: ¥600,000、Indirect Cost: ¥180,000); Fiscal Year 2018: ¥780,000 (Direct Cost: ¥600,000、Indirect Cost: ¥180,000); Fiscal Year 2017: ¥780,000 (Direct Cost: ¥600,000、Indirect Cost: ¥180,000)
• Keywords: 対称群 / ヤング図形 / 既約指標 / ランダム行列 / Weingarten calculus / ランダムヤング図形 / ランダム分割 / 極限形 / 射影表現 / 組合せ論 / 表現論 / 確率論

Outline of Final Research Achievements

As a study on random matrices, especially on the Weingarten calculus, we have investigated the distribution of its partial traces for uniformly distributed random Hermitian matrices on tensor products. For random partitions, we have obtained Kerov polynomial representations and Stanley character formulas for the normalized spin irreducible characters of the symmetric group, and used them to obtain the law of large numbers and the central limit theorem for random shifted Young diagrams.

Academic Significance and Societal Importance of the Research Achievements

ランダム行列論はその名の通りランダムに振る舞う行列の集団を扱う分野で、数理物理学や量子情報理論を始め広い範囲に応用されている。また、その離散への類似物として、整数の分割をランダムに扱う、ランダム分割の研究がある。本研究はこれらについて具体的で視覚的に分かりやすい結果を得た。

Research Products

• [Int'l Joint Research] ニューヨーク大学(米国)
• [Int'l Joint Research] ポーランド科学アカデミー(ポーランド)
• [Int'l Joint Research] ポーランド科学アカデミー(ポーランド)
• [Int'l Joint Research] カリフォルニア大学サンディエゴ校(米国)
• [Int'l Joint Research] ポーランド科学アカデミー(ポーランド)
• [Int'l Joint Research] ポーランド科学アカデミー(ポーランド)
• [Int'l Joint Research] ポーランド科学アカデミー(ポーランド)
• [Int'l Joint Research] カリフォルニア大学(米国)
• [Journal Article] Moments of random quantum marginals via Weingarten calculus (2023)
  • Author(s): S. Matsumoto and C. McSwiggen
  • Journal Title: International Mathematics Research Notices Volume: -
  • Peer Reviewed / Int'l Joint Research
• [Journal Article] 長方形ヤング図形に対応する対称群指標の新しい表示 (2022)
  • Author(s): 松本 詔
  • Journal Title: 数理解析研究所講究録 Volume: 2216 Pages: 64-81
  • Open Access
• [Journal Article] Symmetric group characters of almost square shape (2022)
  • Author(s): S. Matsumoto and P. Sniady
  • Journal Title: Algebraic Combinatorics Volume: 5 Issue: 4 Pages: 771-784
  • DOI: 10.5802/alco.247
  • Peer Reviewed / Open Access / Int'l Joint Research
• [Journal Article] The Weingarten Calculus (2022)
  • Author(s): B. Collins Benoit, S. Matsumoto Sho, J. Novak
  • Journal Title: Notices of the American Mathematical Society Volume: 69 Issue: 05 Pages: 734-745
  • DOI: 10.1090/noti2474
  • Peer Reviewed / Open Access / Int'l Joint Research
• [Journal Article] The Weingarten calculus (2022)
  • Author(s): Benoit Collins, Sho Matsumoto, Jonathan Novak
  • Journal Title: Notices Of The American Mathematical Society Volume: -
  • Peer Reviewed / Int'l Joint Research
• [Journal Article] 対称群のスピン表現に対するStanley指標公式 (2020)
  • Author(s): 松本詔
  • Journal Title: 数理解析研究所講究録 Volume: 2161 Pages: 58-72
  • Open Access
• [Journal Article] Linear versus spin: representation theory of the symmetric groups (2020)
  • Author(s): Sho Matsumoto, Piotr Sniady
  • Journal Title: Algebraic Combinatorics Volume: 3 Issue: 1 Pages: 249-280
  • DOI: 10.5802/alco.92
  • Peer Reviewed / Open Access / Int'l Joint Research
• [Journal Article] Random strict partitions and random shifted tableaux (2020)
  • Author(s): Sho Matsumoto, Piotr Sniady
  • Journal Title: Selecta Mathematica Volume: 26
  • Peer Reviewed / Open Access / Int'l Joint Research
• [Journal Article] A Spin Analogue of Kerov Polynomials (2018)
  • Author(s): Sho Matsumoto
  • Journal Title: Symmetry, Integrability and Geometry: Methods and Applications Volume: 14 Pages: 53-53
  • DOI: 10.3842/sigma.2018.053
  • Peer Reviewed / Open Access
• [Journal Article] A moment method for invariant ensembles (2018)
  • Author(s): Sho Matsumoto and Jonathan Novak
  • Journal Title: Electronic Research Announcements Volume: 25 Issue: 0 Pages: 60-71
  • DOI: 10.3934/era.2018.25.007
  • Peer Reviewed / Int'l Joint Research
• [Journal Article] Polynomiality of shifted Plancherel averages and content evaluations (2017)
  • Author(s): Sho Matsumoto
  • Journal Title: Annales Mathematiques Blaise Pascal Volume: 24 Issue: 1 Pages: 55-82
  • DOI: 10.5802/ambp.364
  • Peer Reviewed / Open Access
• [Journal Article] Weingarten calculus via orthogonality relations: new applications (2017)
  • Author(s): Benoit Collins and Sho Matsumoto
  • Journal Title: ALEA, Lat. Am. J. Probab. Math. Stat. Volume: 14 Pages: 631-656
  • Peer Reviewed / Open Access
• [Presentation] Moments of partial traces for random Hermitian matrices and Weingarten calculus (2023)
  • Author(s): Sho Matsumoto
  • Organizer: Seminar Bielefeld - Melbourne Random Matrices
  • Int'l Joint Research
• [Presentation] 対称群の正規化指標に関する幾つかの結果とスピン類似 (2022)
  • Author(s): 松本 詔
  • Organizer: Non-Commutative Probability and Related Topics 2022
• [Presentation] 長方形ヤング図形に対応する対称群指標の新しい表示 (2021)
  • Author(s): 松本詔
  • Organizer: RIMS共同研究(公開型)「組合せ論的表現論および関連分野との連携」
• [Presentation] ほぼ正方形のヤング図形に対応する対称群の正規化指標 (2021)
  • Author(s): 松本詔
  • Organizer: 第2回 分布族・離散集合の幾何学とその周辺
• [Presentation] 対称群のスピン指標のStanley指標公式 (2020)
  • Author(s): 松本詔
  • Organizer: JCCA2020-DMIA2020-SGT9
  • Invited
• [Presentation] Kerov多項式とStanley指標公式 (2019)
  • Author(s): 松本詔
  • Organizer: 可換確率論と非可換確率論の相互作用準備会
• [Presentation] Kerov polynomials and Stanley formulas for spin symmetric groups (2019)
  • Author(s): Sho Matsumoto
  • Organizer: Interactions between commutative and non-commutative probability
  • Int'l Joint Research
• [Presentation] 対称群のStanley指標公式とmapの数え上げ (2019)
  • Author(s): 松本詔
  • Organizer: Study Meeting on Graph and Curvature
• [Presentation] 対称群のスピン表現に対するStanley指標公式 (2019)
  • Author(s): 松本詔
  • Organizer: RIMS共同研究（公開型）「表現論とその組合せ的側面」
  • Int'l Joint Research
• [Presentation] Weingarten calculus for random matrices associated with the compact symmetric space of class D III (2019)
  • Author(s): Sho Matsumoto
  • Organizer: Workshop on "Random Matrices and Related Topics"
  • Int'l Joint Research
• [Presentation] Weingarten calculus and counting paths on Weingarten graphs (2018)
  • Author(s): Sho Matsumoto
  • Organizer: Random matrices and their applications
  • Int'l Joint Research / Invited
• [Presentation] 直交群の Weingarten calculus と Weingarten グラフ (2018)
  • Author(s): 松本詔
  • Organizer: 2018年度（第57回）実函数論・函数解析学合同シンポジウム
  • Invited
• [Presentation] 対称群のスピン表現に対するKerov多項式 (2018)
  • Author(s): 松本詔
  • Organizer: RIMS共同研究（公開型）「組合せ的表現論の諸相」
  • Int'l Joint Research
• [Presentation] Weingarten calculus for Haar distributed matrices and Wishart matrices (2017)
  • Author(s): 松本詔
  • Organizer: 確率・統計・行列ワークショップ松本2017
• [Remarks] 松本詔のホームページ
  • URL: https://sites.google.com/site/shomatsumotomath1/
• [Remarks] 松本詔のホームページ
  • URL: https://sites.google.com/site/shomatsumotomath1/home
• [Funded Workshop] Workshop on "Random Matrices and Related Topics" (2019)