Rationality of elgenvalues of Cartan matrices in finite groups

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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.