DOUBLE GROUPS AND PROJECTIVE REPRESENTATIONS .1. GENERAL-THEORY

The spinor representations of the full rotation group are double-valued and can be made into projective representations by a method due to Weyl, that depends on the existence of two classes of homotopy in the parameter space of the group. It is shown how this method can be applied to finite point groups, thus avoiding the construction of their double groups, which entail inherent uncertainties. The use of the projective representations also simplifies the work considerably, since half the number of operations is used than in the double group method. Precise techniques are given for the construction of projective factor systems for the spinor representations of all point groups. In § 5 of this paper a general set of theorems is obtained to give the properties of the characters of the projective representations of any finite group. These theorems are considerably simpler than those hitherto available in the literature. From them, and from the theory previously given for point groups, it is shown that the factor systems defined in this paper offer the remarkable advantage that the characters of the corresponding projective representations are class functions for all point groups. Although the present treatment bypasses the use of double groups, it is applied to them in order to clarify various problems that arise and a proof of the Opechowski theorem is given which does not require, as in the original case, the definition of the double group as a group of matrices. © 1979 Taylor & Francis Ltd.