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Rationality of elgenvalues of Cartan matrices in finite groups

有限群中Cartan矩阵的elgen值的有理性

基本信息

批准号：
14540012
负责人：
WADA Tomoyuki
金额：
$ 1.92万
依托单位：
Tokyo University 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：
2002
资助国家：
日本
起止时间：
2002 至 2003
项目状态：
已结题

项目摘要

Let G be a finite group and Fbe an algebraically closed field of characteristic p>0. Let FG be the group algebra and let B be a block of FG with defect group D. Let C_B be the Cartan matrix of Band let ρ (B)be the Frobenius-Perron eigenvalue of C_B. It is the purpose of this research to investigate when the sets of eigenvalues and elementary divisors of C_B coincide. We have the following results and conjecture.1.Bis of finite type (i.e.Dis cyclic), or of tame type (i.e.p=2 and Dis dihedral, generalized quaternion or semidihedral), then the following are equivalent. If G is p-solvable, then The following (1) and (2) are equivalent.(1)The sets of eigenvalues and elementary divisors of C_B coincide.(2)ρ(B)=|D|.(3)ρ(B)is an integer.Furthermore, in this case, B and its Brauer correspondent b are Morita equivalent, if B is of finite type or tame type.2.Conjecture Let E be the set of elementary divisors of C_B and let f_B(x) be the characteristic polynomial of C_B. Let f_B(x)=f_1(x)f_2(x)・・・f_r(x) be the Z-irreducible decomposition. Let R_i be the set of roots of f_i(x). Then there is a disjoint decomposion E=E_1 U・・・UE_r such that the following (i),(ii),(iii) hold.(i)|R_i|= |E_i| for all i.(ii)The product of eigenvalues and elementary divisors in R_i and E_i coinside.(iii)Let ρ(B) is in R_1. Then |D| is in E_1.3.We have the following affirmative result for Conjecture. If B is of finite type and the number of irreducible B-modules≦5,or B is of tame type then Conjecture is true. Furthermore, in this case, if ρ(B,)is a root of f_1(x), then deg f_1≧deg f_i for all i. Moreover, we have verified Conjecture being true for non abelian finite simple groups whose Cartan matrices are known.

设G是有限群，F是特征p> 0的代数闭域.设FG是群代数，B是FG的一个带亏群D的块.设C_B为Band的Cartan矩阵，ρ（B）为C_B的Frobenius-Perron特征值.本文的目的是研究C_B的特征值集与初等因子集何时重合。我们得到了如下结果和定理：1.如果B是有限型的（即D是循环的），或B是驯服型的（即p = 2且D是二面体、广义四元数或半二面体的），则下列是等价的。如果G是p-可解的，则下列（1）和（2）是等价的。（1）C_B的特征值集与初等因子集重合。(2)ρ（B）=| D|. (3)ρ（B）为整数，且当B为有限型或驯服型时，B与其Brauer对应B为Morita等价. 2.猜想设E为C_B的初等因子集，f_B（x）为C_B的特征多项式.设f_B（x）= f_1（x）f_2（x）··f_r（x）是Z-不可约分解.设R_i是f_i（x）的根的集合。然后存在不相交的分解E = E_1 U···UE_r，使得以下（i）、（ii）、（iii）成立。(i)|R_i| =| E_i|为了所有我。（ii）R_i和E_i中特征值与初等因子的乘积。（iii）设ρ（B）在R_1中。然后|D|在E_1中。3.我们对猜想有如下肯定结果。如果B是有限型的，且不可约B-模的个数≥ 5，或B是驯服型的，则猜想成立。此外，在这种情况下，如果ρ（B，）是f_1（x）的根，则对于所有i，deg f_1 <$deg f_i。此外，我们还证明了对于Cartan矩阵已知的非交换有限单群，猜想成立。