On isomorphisms of a Brauer character ring onto another (Q1375737)

Let \(G\) and \(H\) be finite groups and let \(\overline Z\) be the ring of all algebraic integers in the field of complex numbers. Let \(\lambda\colon\overline ZR(G)\to\overline ZR(H)\) be a \(\overline Z\)-algebra isomorphism, where \(\overline ZR(G)\) is the \(\overline Z\)-algebra spanned by the set of irreducible complex characters of \(G\), and let \(A\) be the matrix of \(\lambda\) with entries in \(\overline Z\). The author gives conditions in terms of \(A\) and of the Cartan matrices of \(G\) and \(H\) under which the Brauer character tables of \(G\) and \(H\) are the same.