Rationality of elgenvalues of Cartan matrices in finite groups

让G成为一个有限的群体，让G成为一个代数上封闭的特征p>0的领域。Let FG be the group algebra，let B be a block of FG with defect group D.让C_B是带的Cartan矩阵让ρ （B）是C_B的弗罗贝尼乌斯-珀龙的特征值。当C_B硬币的一组eigenvalues和基本divisors进行调查时，这是本研究的目的。我们有以下结果和假设：1.最小类型（i. e. Dis cyclic）或tame类型（i.e. p =2和Dis dihedral，一般四分之一或半），然后后续是相等的。如果G是可负担的，那么后面的（1）和（2）是相等的。（1）C_B硬币的原始值和基本分区的集合。（b）（B）=| D. D|.（3）ρ（B）is an integer.Furthermore，在这种情况下， B及其Brauer correspondent b是Morita equivalent，如果B是有限类型或tame类型.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。让R_i成为f_i（x）的根集合。Then there is a disjoint decomposion E=E_1 U···UE_r such that the following （i）、（ii）、（iii） hold。（一）|R_i| = |E_i|所有的I.（ii）R_i和E_i coinside中的eigenvalues和elementary divisors的产品。（iii）Let ρ（B） is in R_1。Then（Then）|D. D|是在E_1.3.我们有以下的同情心结果。如果B是有限类型，而不可减少的B-模块的数目≤ 5，或者B是匹配类型，那么这种情况是真的。在这种情况下，如果ρ（B，）是f_1（x）的根，那么f_1 ≥ f_i为所有i。Moreover，我们已经证实了一个事实，对于非阿拉伯裔少数几个简单的群体，谁知道的Cartan矩阵。

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.