Various studies of homogeneous open convex cones and homogeneous Siegel domains

• Project/Area Number: 15K04901
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Basic analysis
• Research Institution: Kyushu University
• Principal Investigator: Nomura Takaaki 九州大学, 数理学研究院, 教授 (30135511)
• Research Collaborator: ISHI Hideyuki
• Project Period (FY): 2015-04-01 – 2019-03-31
• Project Status: Completed (Fiscal Year 2018)
• Budget Amount: ¥4,680,000 (Direct Cost: ¥3,600,000、Indirect Cost: ¥1,080,000); Fiscal Year 2018: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000); Fiscal Year 2017: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000); Fiscal Year 2016: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000); Fiscal Year 2015: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000)
• Keywords: 等質開凸錐 / 正定値対称行列 / 左対称代数 / 向き付けグラフ / ホブソンの公式 / クラン / 球面調和函数 / リー群の表現 / 双対ペア / Hobsonの公式 / 等質錐 / 等質錐の行列実現 / 基本相対不変式 / 対称錐の特徴付け / 調和解析 / 等質空間 / リー群 / 概均質ベクトル空間

Outline of Final Research Achievements

(1) Realization of homogeneous open convex cones: With the help of weighted oriented graphs, I realized any homogeneous open convex cones by stapling subcones of positive-definite real symmetric matrices. Based on this result, we got a characterization of irreducible symmetric cones by means of the degrees of the basic relative invariants.
(2) Modernization of the proof of Hobson's formula: This formula is found by Hobson in 1894, but his proof is too technical to understand. I have succeeded in giving a clearer proof by adopting a Lie-algebraic method, and published an academic paper.

Academic Significance and Societal Importance of the Research Achievements

等質開凸錐の理論は Vinberg に始まり，Koecher，Dorfmeister，Rothaus 等の研究者によって，理論的な整備がなされてきた．しかしながら，どの論文も難しく，とくに非専門家にとっては，等質開凸錐は明示的に記述される基本的な領域であっても，容易には近づけない対象であった．本研究成果はこの現状を打開し，等質開凸錐を応用する研究者に容易なアプローチの道を開くとともに，群作用のない等質開凸錐への一般化を図ることができると期待されるものである．
オイラー作用素とラプラシアンとの交換関係を用いたHobsonの公式の証明は，すでにDunkl解析への応用がななされている．

Research Products

• [Journal Article] A proof of Hobson's formula with the Euler operator (2019)
  • Author(s): Takaaki Nomura
  • Journal Title: Kyushu J. Math. Volume: 73
  • NAID: 130007521574
  • Peer Reviewed / Open Access
• [Journal Article] A PROOF OF HOBSON'S FORMULA WITH THE EULER OPERATOR (2018)
  • Author(s): NOMURA Takaaki
  • Journal Title: Kyushu Journal of Mathematics Volume: 72 Issue: 2 Pages: 423-427
  • DOI: 10.2206/kyushujm.72.423
  • NAID: 130007521574
  • Peer Reviewed / Open Access / Int'l Joint Research
• [Journal Article] A CHARACTERIZATION OF SYMMETRIC CONES BY THE DEGREES OF BASIC RELATIVE INVARIANTS (2016)
  • Author(s): T. Yamasaki and T. Nomura
  • Journal Title: Kyushu Journal of Mathematics Volume: 70 Issue: 2 Pages: 237-257
  • DOI: 10.2206/kyushujm.70.237
  • NAID: 130005421167
  • ISSN: 1340-6116, 1883-2032
  • Peer Reviewed / Open Access / Acknowledgement Compliant
• [Journal Article] A characterization of symmetric cones by the degrees of basic relative invariants (2016)
  • Author(s): T. Yamasaki and T. Nomura
  • Journal Title: Kyushu J. Math. Volume: 70
  • NAID: 130005421167
  • Peer Reviewed / Open Access / Acknowledgement Compliant
• [Journal Article] Realization of homogeneous cones through oriented graphs (2015)
  • Author(s): 山崎貴史，野村隆昭
  • Journal Title: 数理解析研究所講究録 Volume: 1977
  • NAID: 130005076150
  • Open Access / Acknowledgement Compliant
• [Presentation] Minimum size matrix realization of a homogeneous cone (2019)
  • Author(s): Takaaki Nomura
  • Organizer: Bolyai Institute (University of Szeged), Analysis Seminar
• [Presentation] Hobson の公式とその応用 (2018)
  • Author(s): 野村隆昭
  • Organizer: 2017年度表現論ワークショップ
• [Presentation] 等質開凸錐についての最近の研究成果 (2018)
  • Author(s): 野村隆昭
  • Organizer: 行列解析の展開・２
  • Invited
• [Presentation] Determinantal expressions of Bernoulli numbers (2017)
  • Author(s): 野村隆昭
  • Organizer: 表現論ワークショップ
  • Place of Presentation: 鳥取市
  • Year and Date: 2017-01-07
• [Presentation] Introductory talk regarding Jordan algebras and symmetric cones (2017)
  • Author(s): Takaaki Nomura
  • Organizer: Order Structures, Jordan Algebras, and Geometry
  • Int'l Joint Research / Invited
• [Presentation] Homogeneous open convex cones : recent results (2017)
  • Author(s): Takaaki Nomura
  • Organizer: Mathematical Methods of Modern Statistics
  • Int'l Joint Research / Invited
• [Presentation] Hobsonの公式からHermite-Weber変換を経て (O(n), sl(2))-duality へ (2017)
  • Author(s): 野村隆昭
  • Organizer: 名古屋大学調和解析セミナー
• [Presentation] Realization of homogeneous open convex cones (2017)
  • Author(s): Takaaki Nomura
  • Organizer: Seminar on Representation Theory and Number Theory
  • Place of Presentation: National University of Singapore
• [Presentation] Optimal Matrix Realization of Homogeneous Cones (2016)
  • Author(s): Takaaki Nomura
  • Organizer: Geometry, Representation Theory, and Differential Equation
  • Place of Presentation: 九州大学
  • Year and Date: 2016-02-16
  • Int'l Joint Research
• [Presentation] 向き付けグラフを介して等質開凸錐を具現化する (2015)
  • Author(s): 山崎貴史，野村隆昭
  • Organizer: 日本数学会秋季総合分科会
  • Place of Presentation: 京都産業大学
  • Year and Date: 2015-09-13
• [Presentation] Realizing Homogeneous Cones through Oriented Graphs (2015)
  • Author(s): Takaaki Nomura
  • Organizer: The Eighth Congress of Romanian Mathematicians
  • Place of Presentation: UNIVERSITATEA ALEXANDRU IOAN CUZA din Iasi (Romania)
  • Year and Date: 2015-06-26
  • Int'l Joint Research
• [Presentation] Realization of Homogeneous Cones through Oriented Graphs (2015)
  • Author(s): 山崎貴史，野村隆昭
  • Organizer: 2015年度RIMS研究集会 表現論および関連する調和解析と微分方程式
  • Place of Presentation: 京都大学数理解析研究所
  • Year and Date: 2015-06-23
• [Book] 球面調和函数と群の表現 (2018)
  • Author(s): 野村隆昭
  • Total Pages: 372
  • Publisher: 日本評論社
  • ISBN: 9784535798182
• [Book] Geometric and Harmonic Analysis on Homogeneous Spaces and Applications (2017)
  • Author(s): Ali Baklouti and Takaaki Nomura (Eds.)
  • Total Pages: 234
  • Publisher: Springer
  • ISBN: 9783319651804
• [Book] 複素関数論講義 (2016)
  • Author(s): 野村隆昭
  • Total Pages: 290
  • Publisher: 共立出版社
• [Remarks] Takaaki Nomura's home page
  • URL: http://www2.math.kyushu-u.ac.jp/~tnomura/
• [Remarks] Takaaki Nomura's Home Page
  • URL: http://www2.math.kyushu-u.ac.jp/~tnomura/
• [Remarks]
  • URL: http://www2.math.kyushu-u.ac.jp/~tnomura/
• [Funded Workshop] 5th Tunisian-Japanese Conference: Geometric and Harmonic Analysis on Homogeneous Spaces and Applications (2017)
• [Funded Workshop] 4th Tunisian-Japanese Conference (2015)
  • Place of Presentation: Monastir (Tunisia)
  • Year and Date: 2015-12-18