Eigenvalues of Cartan matrices of principal 2-blocks with abelian defect groups

Let G be a finite group with an abelian Sylow 2-subgroup P. Let CB be the Cartan matrix of the principal 2-block B of G. We show that the Frobenius–Perron eigenvalue ρ(B) of CB is a rational integer if and only if B and its Brauer correspondent block b of NG(P) are Morita equivalent by using a classification of finite simple groups with an abelian Sylow 2-subgroup. In this case, we can take the Brauer character table Φb of b as a unimodular eigenvector matrix UB of CB over a complete discrete valuation ring R.