Eigenvalues of Cartan matrices of principal 3-blocks of finite groups with abelian Sylow 3-subgroups

AbstractLet A be the principal 3-block of a finite group G with an abelian Sylow 3-subgroup P. Let CA be the Cartan matrix of A, and we denote by ρ(CA) the unique largest eigenvalue of CA. The value ρ(CA) is called the Frobenius–Perron eigenvalue of CA. We shall prove that ρ(CA) is a rational number if and only if A and the principal 3-block of NG(P) are Morita equivalent. This generalizes earlier Wada's theorem in 2007, where he proves it only for the case that the order of P is nine, while we prove it for the case that P is an arbitrary finite abelian 3-group. The result presented here uses the classification of finite simple groups