Projective group representations and centralizers: Character theory. (English) Zbl 0841.20015

Delvaux, L.; Nauwelaerts, E.

J. Algebra 168, No. 1, 314-339 (1994).

In [ibid. 157, No. 1, 63-79 (1993; Zbl 0785.20009)] we studied the relationship between indecomposable modules over the twisted group rings \(R*_\alpha G\), \(R*_\alpha H\) and the centralizer \(S\) of \(R*_\alpha H\) in \(R*_\alpha G\), where \(R\) is a commutative ring (satisfying suitable conditions), \(G\) is a finite group with \(|G|^{-1}\in R\) and \(H<G\). These results are reviewed and sharpened in Section 1 and the corresponding character theory is developed in Section 2. This work can also be viewed as an extension of Clifford theory (dealing with normal subgroups). In Section 2 we present one of Clifford’s theorems for indecomposable modules over twisted group rings.
Furthermore, we derive orthogonality relations for trace functions on \(S\) and we express primitive central idempotents of \(S\) in terms of trace functions (Section 3). These results are presented in a more general context, namely for Frobenius algebras over rings. Section 4 deals with indecomposable modules and trace functions for algebras of the form \(\varepsilon A\varepsilon\), \(\varepsilon\) being an idempotent. We also focus on the relation between \(S\) and \(\varepsilon(R*_\alpha G)\varepsilon\), where \(\varepsilon\) is a primitive idempotent of \(R*_\alpha H\).

MSC:

20C25	Projective representations and multipliers
20C15	Ordinary representations and characters
20C05	Group rings of finite groups and their modules (group-theoretic aspects)
16L60	Quasi-Frobenius rings

Keywords:

indecomposable modules; twisted group rings; centralizers; finite groups; Clifford theory; orthogonality relations; trace functions; primitive central idempotents; Frobenius algebras