Research on p-adic properties of the numbers of permutation representations

• Project/Area Number: 22540004
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Single-year Grants
• Section: 一般
• Research Field: Algebra
• Research Institution: Muroran Institute of Technology
• Principal Investigator: TAKEGAHARA Yugen 室蘭工業大学, 工学(系)研究科(研究院), 教授 (10211351)
• Project Period (FY): 2010-04-01 – 2014-03-31
• Project Status: Completed (Fiscal Year 2013)
• Budget Amount: ¥4,290,000 (Direct Cost: ¥3,300,000、Indirect Cost: ¥990,000); Fiscal Year 2013: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000); Fiscal Year 2012: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000); Fiscal Year 2011: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000); Fiscal Year 2010: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000)
• Keywords: 準同型 / 対称群 / 交代群 / 環積 / p進的性質 / ｐ進的性質 / 指数型母関数 / 母関数 / p進解析関数 / 有限アーベルp群 / p進整数 / 巡回群 / インボリューション / 有限アーベル群 / 自由積

Research Abstract

There are three results. 1) p-adic properties of the number of homomorphisms from a finite abelian p-group to symmetric groups were obtained. In particular, the exponent of p of the decomposition of such numbers into prime factors was made clear. There are 6 different types of the results. 2) The exponent of 2 in the decomposition of one plus the number of involutions in alternating groups or wreath products of a cyclic group of order 2 by alternating groups groups into prime factors could be described as 2-adic integers. 3)p-adic properties of the number of homomorphisms from the direct product of two cyclic p-groups to wreath product of a cyclic p-group by symmetric groups were obtained. The results are closely related to that for the number of homomorphisms to symmetric groups.

Research Products

• [Journal Article] Induction formulae for Mackey functors with applications to representations of the twisted quantum double of a finite group (2014)
  • Author(s): Y. Takegahara
  • Journal Title: J. Algebra Volume: 410 Pages: 85-147
  • DOI: 10.1016/j.jalgebra.2014.03.017
  • NAID: 120005754530
  • Peer Reviewed
• [Journal Article] On a theorem of P. Hall (2013)
  • Author(s): T. Asai, N. Chigira, T. Niwasaki, and Y. Takegahara
  • Journal Title: J. Group Theory Volume: 16 Pages: 69-80
  • Peer Reviewed
• [Journal Article] On a Theorem of P. Hall (2013)
  • Author(s): Tsunenobu Asai
  • Journal Title: Journal of Group Theory Volume: 16 Pages: 69-80
  • Peer Reviewed
• [Journal Article] Generalizations of Burnside ring and their applications (2011)
  • Author(s): 竹ヶ原裕元
  • Journal Title: 数理解析研究所講究録 Volume: 1756 Pages: 41-50
• [Journal Article] Multiple Burnside rings and Brauer induction formulae (2010)
  • Author(s): Y. Takegahara
  • Journal Title: J. Algebra Volume: 324 Issue: 7 Pages: 1656-1686
  • DOI: 10.1016/j.jalgebra.2010.05.021
  • Peer Reviewed
• [Journal Article] Multiple Burnside ring and Brauer induction formulae (2010)
  • Author(s): Yugen Takegahara
  • Journal Title: J.Algebra Volume: 32 Pages: 1656-1686
  • Peer Reviewed
• [Presentation] 2-adic properties of the number of solutions of x^m=1 in the alternating group A_n (2014)
  • Author(s): 竹ヶ原裕元
  • Organizer: 有限群とその表現,頂点作用素代数, 代数的組合わせ論の研究
  • Place of Presentation: 京都大学数理解析研究所
• [Presentation] The crossed and monomial Burnside rings (2013)
  • Author(s): 竹ヶ原裕元
  • Organizer: 第58回代数学シンポジウム
  • Place of Presentation: 広島大学
• [Presentation] バーンサイド環の一般化とその応用 (2012)
  • Author(s): 竹ヶ原裕元
  • Organizer: 有限群とその表現, 頂点作用素代数,組合せ論の研究
  • Place of Presentation: 京都大学数理解析研究所(招待講演)
  • Year and Date: 2012-03-05
• [Presentation] バーンサイド環の一般化とその応用有限群とその表現 (2012)
  • Author(s): 竹ヶ原裕元
  • Organizer: 頂点作用素代数, 組合せ論の研究
  • Place of Presentation: 京都大学数理解析研究所
• [Presentation] Generalizations of Burnside ring and their applications (2010)
  • Author(s): Yugen Takegahara
  • Organizer: Vertex Operator Algebras, Finite Groups and Combinatorics
  • Place of Presentation: 京都大学数理解析研究所
  • Year and Date: 2010-12-14
• [Presentation] Generalizations of Burnside ring and their applications (2010)
  • Author(s): 竹ヶ原裕元
  • Organizer: 頂点作用素代数・有限群・組合せ論の研究
  • Place of Presentation: 京都大学数理解析研究所
• [Presentation] 2-adic properties for the number of solutions of x^m=1 in the alternating group A_n
  • Author(s): 竹ヶ原 裕元
  • Organizer: RIMS研究集会 有限群とその表現、頂点作用素代数、代数的組合わせ論の研究
  • Place of Presentation: 京都大学数理科学研究所（京都府京都市）
  • Invited