Orthogonal representations of finite groups and group rings. (Q2762224)

In this dissertation work (Habilitationsschrift) orthogonal representations of finite groups and group rings are considered. It consists of five chapters. In chapter 1 an algebraic introduction is given. In chapter 2 a quantitative version of Frobenius reciprocity for orthogonal modules is proved. Then it is applied to deduce recursion formulas for the rational class of some irreducible orthogonal modules. In chapter 3, for finite groups \(G\) several methods are developed to describe isometry classes of \(G\)-invariant forms on irreducible \(\mathbb{Q} G\)-modules. As examples, all rational orthogonal representations of groups \(\text{SL}_2(5)\), \(M_{11}\) and \(\text{Sp}_4(3)\) are classified, as well as for cyclic groups. In chapter 4 group rings over integer \(p\)-adic rings are studied as orders with involutions. In chapter 5 methods are developed to describe symmetric orders with involutions, in particular group rings, as orders with involutions up to Morita equivalence. Finally, in the last chapter the group rings \(\mathbb{Z}_p[\zeta_{p^f-1}]\text{SL}_2(p^f)\) are calculated (nearly) up to Morita equivalence.NEWLINENEWLINE Note that most of the results can be found also in journal articles of the author [J. Algebra 225, No. 1, 250-260 (2000; Zbl 0954.20005), Exp. Math. 9, No. 4, 623-629 (2000; Zbl 0978.20004), J. Algebra 210, No. 2, 593-613 (1998; Zbl 0923.20003), J. Reine Angew. Math. 528, 183-200 (2000; Zbl 1043.20009), J. Algebra 230, No. 2, 424-454 (2000; Zbl 1043.20008)].