Projective group representations and centralizers: Character theory.

*(English)*Zbl 0841.20015In [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 \(R*_\alpha H\).

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 |