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A STUDY OF THE CARTAN MATRICES OF FINITE GROUPS

有限群嘉当矩阵的研究

基本信息

批准号：
09640015
负责人：
WADA Tomoyuki
金额：
$ 1.6万
依托单位：
Tokyo University of Agriculture & Technology
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
1997
资助国家：
日本
起止时间：
1997 至 1999
项目状态：
已结题

项目摘要

Let G be a finite group, FG the group algebra of G over an algebraically closed field F of characteristic p > 0, and B a block of FG. Let CィイD2BィエD2 be the Cartan matrix of B and p(B) the Perron-Frobenius eigenvalue (i.e. the largest one) of CィイD2BィエD2. Let k(B), l(B) be the number of ordinary and modular (i.e. over F) irreducible characters, respectively.We have found k(B) 【less than or equal】 p(B)l(B) holds for any block B of G in [1], further we posed a conjecture that k(B) 【less than or equal】 p(B) for any p-solvable groups. This is a stronger assertion than Brauer's famous k(B)-conjecture for p-solvable groups. In [2] we completed to calculate the Cartan matrix of a certain class of finite solvable groups, which have many 0 entries. We also verified that k(B) 【less than or equal】 p(B) in this case. In [3] we investigated on the fundamental question when eigenvalus and elementary divisors of CィイD2BィエD2 coincide. We have had a striking conjecture that p(B) is rational if and only if B is Morita equivalent to the Brauer correspondent b of NィイD2GィエD2(D). In [4] we newly obtained a definition of P-good module and p-good group which is related to eigenvalues of CィイD2BィエD2.[1] T. Wada, A lower bound of the Perron-Frobenius eigenvalue of the Cartan matrix for finite groups. Arch. Math. 73 (1999), 407-413. [2] T. Wada, The Cartan matrix of a certain class of finite solvable groups. (accepted to Osaka Jour. Math.) [3] M. Kiyota, M. Murai and T. Wada, Rationality of eigenvalues of the Cartan matrices of finite groups. (in preperation) [4] A. Hanaki, M. kiyota, M. Murai and T. Wada, P-good modules and p-good groups. (in preperation)

设G是有限群，FG是特征为p&gt；0的代数闭域F上的群代数，B是FG的块。设CィイD2BィエD2是B的Cartan矩阵，p(B)是CィイD2BィエD2的Perron-Frobenius特征值(即最大的一个)。设k(B)，L(B)分别是普通特征标和模(即在F上)不可约特征标的个数，我们在文[1]中发现了k(B)[小于等于]p(B)L(B)对G的任意块B成立，进而提出了对任意p-可解群k(B)[小于等于]p(B)的猜想.这是一个比Brauer关于p-可解群的著名的k(B)猜想更强的断言。在文[2]中，我们完成了一类有许多0项的有限可解群的Cartan矩阵的计算。在这种情况下，我们还证明了k(B)[小于或等于]p(B)。在文[3]中，我们研究了CィイD2BィエD2的本征值与初等因子重合的基本问题。我们有一个惊人的猜想：p(B)是有理的当且仅当B是Morita等价于NィイD2GィエD2(D)的Brauer对应b。在文[4]中，我们得到了与CィイD2BィエD2的特征值有关的P-良模和P-良群的定义。[1]T.Wada，有限群的Cartan矩阵的Perron-Frobenius特征值的下界。拱门。数学课。73(1999)，407-413。[2]T.Wada，某类有限可解群的Cartan矩阵。(接受大阪巡回赛。数学)[3]M.Kiyota，M.Murai和T.Wada，有限群的Cartan矩阵的特征值的合理性。[4]A.Hanaki，M.Kiyota，M.Murai和T.Wada，P-Good模和P-Good群。(在准备中)