Cohomology of finite groups and homotopy theory of classifying space from the view point of representation theory

• Project/Area Number: 24540007
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Algebra
• Research Institution: Saitama University
• Principal Investigator: HIDA Akihiko 埼玉大学, 教育学部, 教授 (50272274)
• Project Period (FY): 2012-04-01 – 2015-03-31
• Project Status: Completed (Fiscal Year 2014)
• Budget Amount: ¥1,430,000 (Direct Cost: ¥1,100,000、Indirect Cost: ¥330,000); Fiscal Year 2014: ¥390,000 (Direct Cost: ¥300,000、Indirect Cost: ¥90,000); Fiscal Year 2013: ¥390,000 (Direct Cost: ¥300,000、Indirect Cost: ¥90,000); Fiscal Year 2012: ¥650,000 (Direct Cost: ¥500,000、Indirect Cost: ¥150,000)
• Keywords: 有限群 / コホモロジー / 分類空間 / Burnside 環 / 表現論 / バーンサイド環

Outline of Final Research Achievements

The double Burnside algebra of a finite group is the algebra with basis corresponding to finite sets such that the group acts from both sides. This algebra contains the information on the splitting of classifying space of finite group in the stable homotopy category. We studied the stable splitting of classifying space of finite groups through the action of double Burnside algebra on the cohomology.
We mainly consider the nonabelian p-group of order p cubed and exponent p. We determined the multiplicity of summands in the stable splitting and cohomology of these summands with coefficient the field of p elements. Moreover, we obtain information on finite groups having this group as a Sylow p-subgroup.

Research Products

• [Journal Article] Representaions of the double Burnside algebra and cohomology of the extraspecial p-group (2014)
  • Author(s): A. HIda, N. Yagita
  • Journal Title: J. Algebra Volume: 409 Pages: 265-319
  • DOI: 10.1016/j.jalgebra.2014.03.021
  • Peer Reviewed
• [Journal Article] Cohomology of the extraspecial p-group and representations of the double Burnside algebra (2014)
  • Author(s): A. Hida
  • Journal Title: 数理解析研究所講究録 Volume: 1872 Pages: 132-139
• [Presentation] Extraspecial p-群の mod p コホモロジーと両側 Burnside 環の作用について (2013)
  • Author(s): 飛田明彦
  • Organizer: 日本数学会秋季総合分科会
  • Place of Presentation: 愛媛大学
• [Presentation] Cohomology of the extraspecial p-group and representations of the double Burnside algebra
  • Author(s): 飛田明彦
  • Organizer: RIMS研究集会 有限群とその表現、頂点作用素代数、代数的組み合わせ論の研究
  • Place of Presentation: 京都大学数理解析研究所
• [Presentation] Extraspecial p-群のコホモロジーへの両側 Burnside 環の作用について
  • Author(s): 飛田明彦
  • Organizer: 日本数学会年会
  • Place of Presentation: 京都大学