Eigenvalues of Cartan matrices of principal 3-blocks of finite groups with Abelian Sylow 3-subgroups. (Q615870)

Let \(A\) be the principal 3-block of a finite group \(G\) with an Abelian Sylow 3-subgroup \(P\), and let \(\varrho\) be the unique largest eigenvalue of the Cartan matrix \(C\) of \(A\). Making use of the classification of finite simple groups, the authors prove that the following assertions are equivalent: (1) \(\varrho\in\mathbb{Z}\); (2) \(\varrho=|P|\); (3) the eigenvalues of \(C\) coincide with the elementary divisors of \(C\); (4) all eigenvalues of \(C\) are in \(\mathbb{Z}\); (5) \(A\) is Morita equivalent to its Brauer correspondent in \(N_G(P)\). This confirms a special case of a conjecture by \textit{T. Wada} [J. Algebra 308, No. 2, 629-640 (2007; Zbl 1119.20014)].