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