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

Let A be the principal 3-block of a finite group G with an abelian Sylow 3-subgroup P. Let C A be the Cartan matrix of A, and we denote by ρ ( C A ) the unique largest eigenvalue of C A . The value ρ ( C A ) is called the Frobenius–Perron eigenvalue of C A . We shall prove that ρ ( C A ) is a rational number if and only if A and the principal 3-block of N G ( 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.