For the solution of Broue's conjecture in representation theory of finite groups

对于有限群表示论中布劳猜想的求解

• 批准号: 14540009
• 负责人: KOSHITANI Shigeo
• 金额: $ 2.56万
• 依托单位: Chiba University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 2002
• 资助国家: 日本
• 起止时间: 2002 至 2004
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-14540009/
• 关键词: finite group; representation theory; modular representation theory; block; abelian defect group; Broue's conjecture; Morita equivalence; irreducible character; デイド予想; 共役類

The head investigator Koshitani has gotten the following 3 results in this research.First, we get a result that if our block is the principal 3-block with elementary abelian defect group of order 9 of an arbitrary finite group, then Broue's conjecture holds anytime. This may be one of the most important results on the conjecture. This was published in the "Journal of Algebra" (Elsevier.).Because of the above result, it should be natural to consider Broue's conjectre as the next step for non-principal 3-block of finite groups with the same elementary abelian defect group of order 9. We did it for other concrete finite groups, say O'Nan's simple group and the Higman-Sims simple group which are sporadic, namely we proved Broue's conjecture is still true for non-principal 3'blocks of these finite simple groups with the same defect group. The result was published in the "Journal of Pure and Applied Algebra" (Elsevier).There are conjectures of Donovan and Puig, both of which are closely related to Broue's abelian defect group conjecture. As the third result in this research, we proved that these two conjectures hold for all principal 3-blocks of finite groups with the same Sylow 3-subgroups which are elementary abelian of order 9. This result was published in a journal "Communications inAlgebra (Dekker)".The head investigator Koshitani announced these 3 results in University of Durham, UK 2002,Torun University, Poland 2003,Banff Science Institute, Canada 2003 and also University of Oxford, UK 2004,as invited talks.We have had another important results due to the other investigators Kitazume and Matsuda, for instance.

首席研究员Koshitani在本研究中得到了以下三个结果：第一，如果我们的块是具有任意有限群的9阶初等交换亏群的主3-块，则Broue猜想在任何时候都成立。这可能是关于该猜想的最重要的结果之一。这篇文章发表在《代数杂志》(Elsevier)上。由于上述结果，对于具有相同9阶初等交换亏群的有限群的非主3-块，应该很自然地考虑Broue猜想作为下一步。我们对其他具体的有限群也是这样做的，比如零星的O‘Nan单群和Higman-Sims单群，即我们证明了Broue猜想对这些具有相同亏群的有限群的非主3’‘块仍然成立。这一结果发表在《理论与应用代数杂志》(Elsevier)上。有Donovan和Puig的猜想，这两个猜想都与Broue的阿贝尔亏群猜想密切相关。作为这项研究的第三个结果，我们证明了这两个猜想对具有相同Sylow 3-子群的所有9阶初等交换的有限群的主3-块都成立。这一结果发表在《通信代数(Dekker)》杂志上。首席研究员Koshitani应邀请在英国杜伦大学2002年、波兰托伦大学2003年、加拿大Banff科学研究所2003年以及英国牛津大学2004年宣布了这三个结果。例如，由于其他研究人员Kitazume和Matsuda的邀请，我们又有了另一个重要的结果。

项目成果

The radical subgroups of the Fischer simple group

费歇尔单群的激进子群

• 发表时间: 2002
• 期刊: Journal of Algebra 255
• 作者: Shigeo Koshitani;Shigeo Koshitani;Shigeo Koshitani;Sohei Nozawa;Shigeo Koshitani;Shigeki Matsuda;Shigeo Koshitani;Shigeki Matsuda;Masaaki Kitazume;Shigeki Matsuda;Shigeo Koshitani;Shigeo Koshitani;Masaaki Kitazume;Shigeo Koshitani;Shigeo Koshitani;Masaaki Kitazume
• 通讯作者: Masaaki Kitazume

Koji Nishida: "Hilbert Coefficients and Buchsbaumness of associated graded rings"Journal of Pure and Applied Algebra. 181・1. 61-74 (2003)

西田浩二：“关联分级环的希尔伯特系数和布克斯鲍姆性”纯粹与应用代数杂志181・1（2003）。

Shigeo Koshitani: "Broue's conjecture for non-principal 3-blocks of finite groups"Journal of Pure and Applied Algebra. 173. 177-211 (2002)

Shigeo Koshitani：“有限群的非主 3 块的布劳猜想”纯粹与应用代数杂志。

Broue's conjecture for non-principal 3-blocks of finite groups

有限群非主 3 块的布劳猜想

• 发表时间: 2002
• 期刊: Journal of Pure and Applied Algebra 173
• 作者: Shigeo Koshitani;Shigeo Koshitani;Shigeo Koshitani;Sohei Nozawa;Shigeo Koshitani;Shigeki Matsuda;Shigeo Koshitani;Shigeki Matsuda;Masaaki Kitazume;Shigeki Matsuda;Shigeo Koshitani;Shigeo Koshitani;Masaaki Kitazume;Shigeo Koshitani;Shigeo Koshitani
• 通讯作者: Shigeo Koshitani

Shigeo Koshitani: "Broue's conjecture holds for principal 3-blocks with elementary abelian defect group of order 9"Journal of Algebra. 248. 575-604 (2002)

Shigeo Koshitani：“Broue 猜想对于具有 9 阶基本阿贝尔缺陷群的主 3 块成立”《代数杂志》。