Study on eigenvalues and elementary divisors of Cartan matrices in finite groups

有限群嘉当矩阵的特征值和初等因数研究

• 批准号: 17540014
• 负责人: WADA Tomoyuki
• 金额: $ 2.09万
• 依托单位: Tokyo University of Agriculture and Technology
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 2005
• 资助国家: 日本
• 起止时间: 2005 至 2007
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-17540014/
• 关键词: finite group; modular representation; block; Cartan matrix; Frobenius-Perron eigenvalue; elementary divisor; Morita equivalent; defect group; フロベニウス・ペロン固有値; モジュラー表現論; 固有ベクトル行列

Let G be a finite group and F be an algebraically closed field of characteristic p>0. Let B be a block of the group algebra FG with defect group D. Let C be the Cartan matrix of B and let ρ be the Frobenius-Perron eigenvalue of C. It is the purpose of this research to investigate the structure of G or B when ρ is an integer. We found from many examples that ρ is an integer if and only if B and its Brauer correspondent b are Morita equivalent. But there is a counter example to this statement when D is not abelian. So, if D is abelian, we conjectured that ρ is an integer if and only if B and b are Morita equivalent. We proved that the conjecture is true in the following.1 When D is cyclic, or p=2 and D is the four group, the conjecture is true.2 When D is elementary abelian 3-group of order 9 and B is the principal 3-block, the conjecture is true.3 When G has an abelian 2-Sylow subgroup and B is the principal 2-block, the conjecture is true.Since Morita equivalence means derived equivalence, our concern that ρ is an integer is in turn deeply related to Broue's abelian defect group conjecture.

设G是有限群，F是特征p>0的代数闭域.设B是群代数FG的块，其亏群为D.设C是B的Cartan矩阵，ρ是C的Frobenius-Perron特征值.本研究的目的是研究当ρ为整数时G或B的结构。通过大量的例子我们发现ρ是整数当且仅当B与它的Brauer对应B是Morita等价的.但是当D不是阿贝尔时，有一个反例。因此，如果D是阿贝尔的，我们证明了ρ是整数当且仅当B和B是Morita等价的.证明了猜想成立的条件是：1当D是循环群，或p=2且D是4-群时，猜想成立; 2当D是9阶初等交换3-群且B是主3-块时，猜想成立; 3当G有交换2-Sylow子群且B是主2-块时，猜想成立;由于Morita等价表示导出等价，我们对ρ是整数的关注又与Broue的交换亏群猜想密切相关。

项目成果

Stability of Frobenius algebras with positive Galois coverings

具有正伽罗瓦覆盖的 Frobenius 代数的稳定性

• 发表时间: 2005
• 期刊: Proceedings of the 37th Symposium on Ring Theory and Representation Theory 37
• 作者: A.Skowronski;K.Yamagata;K.Yamagata
• 通讯作者: K.Yamagata

Proceedings of American Mathematical

美国数学会刊

• 发表时间: 2005
• 期刊: Society 133
• 作者: H.;Fukushima
• 通讯作者: Fukushima

Integrality of eigenvalues of Cartan matrices in finite groups

有限群中Cartan矩阵特征值的完整性

• 发表时间: 2007
• 期刊: Proceedings of the 39th Symposium of Ring Theory and Representation Theory 39
• 作者: J.;Bialkowski;A.;Skowronski;K.;Yamagata;T. Wada
• 通讯作者: T. Wada

Irreducible products of characters of solvable groups

可解群特征的不可约积

• 发表时间: 2007
• 作者: A.Skowronski;K.Yamagata;K.Yamagata;Y.Usami;H.Fukushima;K. Yamagata;M. Kiyota;M. Kiyota;H. Fukushima
• 通讯作者: H. Fukushima

Hall subgroups of M-groups need not be M-groups

M 群的霍尔子群不必是 M 群

• 发表时间: 2005
• 期刊: Proc. Amer Math Soc 133
• 作者: Ohtake;Koichiro;福島 博;Tomoyuki Wada;Koichiro Ohtake;福島 博;Fukushima Hiroshi;Hiroshi Fukushima
• 通讯作者: Hiroshi Fukushima