On isomorphisms of a Brauer character ring onto another. II (Q5945585)

[For part I cf. Tsukuba J. Math. 20, No. 1, 207-212 (1996; Zbl 0891.20006).]NEWLINENEWLINENEWLINEIf \(G\) is a finite group, let \(\overline\mathbb{Z} BR(G)\) denote the ring of \(\overline\mathbb{Z}\)-linear combinations of Brauer characters of \(G\), where \(\overline\mathbb{Z}\) is the ring of all algebraic integers in the field of complex numbers.NEWLINENEWLINENEWLINEAssume that there is an isomorphism \(\lambda\colon\overline\mathbb{Z} BR(G)\to\overline\mathbb{Z} BR(H)\), where \(G\) and \(H\) are finite groups. The author gives a necessary and sufficient condition in terms of the Cartan matrices of \(G\) and \(H\), under which the isomorphism \(\lambda\) preserves a certain inner product. He also gives a condition under which \(\lambda\) is compatible with decomposition into \(p\)-blocks.