I'm not a mathematician. I've only learnt about irreducible representation of finite group, symmetric group and simple Lie group. In fact, I don't know projective representation belong to which part in mathematics (I heard some people say it needs group cohomology). I heard that projective representation of a Lie group is just the representation of its universal covering group and I always use this result but I really don't known its proof. Recently I may need to use the complex and real projective representation of finite group (especially space point group and permutation group), so if anyone could give some direct reference it would be very useful.

(1) Is there some reference of real and complex projective representation of finite group? I may only need the representation in real and complex field and don't need the representation in general field.

(2) Where can I find the proof that projective representation of a Lie group is the representation of its universal covering group?

  • $\begingroup$ It sounds that you are talking about exactly Kleshchev, Alexander. Linear and projective representations of symmetric groups. Vol. 163. Cambridge University Press, 2005. since it focus primarily on the projective representation. But it is hard even for math people.... $\endgroup$ – Henry.L Feb 24 '17 at 23:54
  • $\begingroup$ Curtis & Reiner's encyclopedic book on representation theory has a proof of Schur's theorem about projective representations of finite groups lifting to representations of the covering group -- this is (53.7) in that text. I have an e-copy if you need one, just email me. $\endgroup$ – Nick Gill Feb 27 '17 at 11:50

(1) For finite groups and complex projective representations, the keyword is Schur multiplier. (I'm not aware of a reference for the real case.)

(2) For Lie groups, the analogous theory was developed by Bargmann (1954). As he explains, your guess that all (complex) projective representations arise from linear representations of the universal covering is true when the group is semi-simple, but not in general.


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