The Brauer splitting theorem and projective representations of finite groups over rings

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In the theory of representations of finite groups the projective representations arise naturally when one studies the relations between the representations of the group and representations of certain subgroups. The role played by the group rings in representation theory is taken by the twisted groups rings when one considers projective representations. For representations over algebraically closed fields the theory of Schur multipliers provides a very satisfactory tool that may be used to lift projective representations of a finite group to usual representations of a finite central extension of that group. Furthermore there exists a satisfactory theory of projective characters, block-theory for projective representations, etc. . . . . for which we may refer to Karpilovsky’s recent book, [3]. Since for finite groups, the twisted group rings defined over commutative rings are Azumaya algebras (when a mild condition holds), it is natural to ask for an equivalent of the Brauer splitting theorem for group rings. In this paper we derive this splitting theorem for twisted group rings over commutative rings without nontrivial idempotents. This is obtained by introducing a suitable extension of the Schur multiplier theorem in such a way that this result may be applied to a single representation and over arbitrary fields or rings of the forementioned type. Along the way we get involved in some cocycle manipulations that will lead to the determination of the center of a twisted group ring in terms of ray-classes. The results derived in this paper should be of some use in the study of properties and