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

Abstract

Let G be a finite group with an abelian Sylow 2-subgroup P. Let C B be the Cartan matrix of the principal 2-block B of G. We show that the Frobenius–Perron eigenvalue ρ ( B ) of C B is a rational integer if and only if B and its Brauer correspondent block b of N G ( 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 Φ b of b as a unimodular eigenvector matrix U B of C B over a complete discrete valuation ring R.