On Morita and derived equivalences for blocks of finite groups

2018 Fiscal Year Final Research Report

• Project/Area Number: 26400026
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Algebra
• Research Institution: Tokyo University of Science
• Principal Investigator: Kunugi Naoko 東京理科大学, 理学部第一部数学科, 准教授 (50362306)
• Project Period (FY): 2014-04-01 – 2019-03-31
• Keywords: 有限群 / モジュラー表現 / ブロック / 森田同値 / 導来同値 / 安定同値

Outline of Final Research Achievements

One of the important conjectures in modular representation theory of finite groups is Broue's abelian defect group conjecture. It states that the principal blocks of two finite groups having a common abelian Sylow subgroup and the same p-local structures should be derived equivalent. To solve the conjecture it is important to develope the way of gluing local derived equkvalences to global stable equivalences and of lifting stable equivalences to derived equivalences. It is also important to investigate non-abelian defect group cases. In this research, We obtained a result for Brauer indecomposability of Scott modules with non-abelian vertex. We also obtained a reselt for construction of two-sided tilting complexes for Brauer tree algebras.

Free Research Field

代数学

Academic Significance and Societal Importance of the Research Achievements

有限群のモジュラー表現において，有限群のブロックの導来同値や森田同値での分類は重要な問題である。とくに可換不足群をもつブロックとその局所部分群のブロックの導来同値性を述べたブルエの可換不足群予想や，指定した群を不足群にもつブロックの森田同値類の有限性を述べたドノバン予想は重要である。本研究の成果は，これらの予想の解決に向けた各ステップにおいて役立つような成果であり，重要な成果であると考えられる。