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

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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.