Characters of groups and algebras as special functions and their applications

• Project/Area Number: 24740033
• Research Category: Grant-in-Aid for Young Scientists (B)
• Allocation Type: Multi-year Fund
• Research Field: Algebra
• Research Institution: 防衛大学校(総合教育学群、人文社会科学群、応用科学群、電気情報学群及びシステム工学群)
• Principal Investigator: MIZUKAWA Hiroshi 防衛大学校(総合教育学群、人文社会科学群、応用科学群、電気情報学群及びシステム工, その他部局等, 准教授 (60531762)
• Project Period (FY): 2012-04-01 – 2015-03-31
• Project Status: Completed (Fiscal Year 2014)
• Budget Amount: ¥4,030,000 (Direct Cost: ¥3,100,000、Indirect Cost: ¥930,000); Fiscal Year 2014: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000); Fiscal Year 2013: ¥1,300,000 (Direct Cost: ¥1,000,000、Indirect Cost: ¥300,000); Fiscal Year 2012: ¥1,560,000 (Direct Cost: ¥1,200,000、Indirect Cost: ¥360,000)
• Keywords: 指標の理論 / 帯球関数 / 対称関数 / 分割恒等式 / 直交多項式 / 代数学 / 表現論 / 組合せ論 / 指標 / 確率論 / 有限群の表現論

Outline of Final Research Achievements

In this study, some results are obtained by using the orthogonality of group or algebra characters. First result is related to stochastic models called Poker-dice game and Ehrenfest's urn model. Their ergodic properties and cut-off phenomenons are analyzed by using representation theoretical method. Second result is a formula on certain algebraic relations of Schur functions. Some formulas are proved by the representation of an affine Lie algebra and the theory of the symmetric functions. Furthermore the representation theoretic meaning of these formulas is revealed. The last result is a combinatorial formula of (class) regular partitions. The relation between the product of all parts of the class regular partitions and that of multiplicities are given.

Research Products

• [Journal Article] Schur function identities arising from the basic representation of $A^{(2)}_{2}$ (2014)
  • Author(s): H. Mizukawa, T. Nakajima, R. Seno and H.-F. Yamada
  • Journal Title: Letters in Mathematical Physics Volume: 104 Issue: 10 Pages: 1317-1331
  • DOI: 10.1007/s11005-014-0717-y
  • Peer Reviewed / Acknowledgement Compliant
• [Journal Article] Poker dice game and multivariate Krawtchouk polynomials (2013)
  • Author(s): 水川裕司
  • Journal Title: 数理解析研究所講究録 Volume: 1870 Pages: 16-24
• [Presentation] r-正則分割の組合せ論 (2014)
  • Author(s): 水川裕司，山田裕史
  • Organizer: 組合せ論的表現論と表現論的組合せ論
  • Place of Presentation: 京都大学
  • Year and Date: 2014-10-30
  • Invited
• [Presentation] 分割恒等式と対称群の指標表 (2014)
  • Author(s): 水川裕司，山田裕史
  • Organizer: 日本数学会秋季総合分科会
  • Place of Presentation: 広島大学
  • Year and Date: 2014-09-26
• [Presentation] Schur function identities arising from the basic representation of A^(2)_2 (2013)
  • Author(s): 水川裕司，中島達洋，山田裕史
  • Organizer: 表現論と組合せ論
  • Place of Presentation: 京都大学
• [Presentation] A^(2)_2の基本表現から得られるシューア関数の恒等式 (2013)
  • Author(s): 水川裕司，中島達洋，山田裕史
  • Organizer: 日本数学会秋季総合分科会
  • Place of Presentation: 愛媛大学
• [Presentation] Interactions between Ehrenfest's urns arising from group actions (2013)
  • Author(s): Hiroshi Mizukawa
  • Organizer: International workshop on Noncommutative analysis and its future prospectives
  • Place of Presentation: 北海道生大学
  • Invited
• [Presentation] 環積が作用する確率モデルと多変数 Krawtchouk 多項式 (2012)
  • Author(s): 水川裕司
  • Organizer: 数理解析研究所 表現論セミナー
  • Place of Presentation: 数理解析研究所
  • Invited
• [Presentation] Multivariate Krawtchouk Polynomials (2012)
  • Author(s): Hiroshi Mizukawa
  • Organizer: FoPSAaC
  • Place of Presentation: 名古屋大学
  • Invited
• [Presentation] Poker dice game and multivariate Krawtchouk polynomials (2012)
  • Author(s): 水川裕司
  • Organizer: Combinatorial Representation Theory and Related Topics
  • Place of Presentation: 数理解析研究所
• [Remarks] research gate
  • URL: https://www.researchgate.net/profile/Hiroshi_Mizukawa?ev=hdr_xprf