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Recently I read the chapter 2 of Weinberg's QFT vol1. I learned that in QM we need to study the projective representation of symmetry group instead of representation. It says that a Lie group can have nontrivial projective represention if the Lie group is not simple connected or the Lie algebra has nontrivial center. So for simple Lie group, the projective representation is the representation of universal covering group.

But it only discuss the Lie group, so what's about the projective representation of discrete group like finite group or infinite discrete group? I heard it's related to group cohomology, Schur's multiplier and group extension. So can anyone recommend some textbooks, monographs, reviews and papers that can cover anyone of following topics which I'm interested in:

How to construct all inequivalent irreducible projective representations of Lie group and Lie algebra? How to construct all inequivalent irreducible projective representations of discrete group? How are these related to central extension of group and Lie algebra ? How to construct all central extension of a group or Lie algebra? How is projective representation related to group cohomology? How to compute group cohomology? Is there some handbooks or list of group cohomology of common groups like $S_n$, point group, space group, braiding group, simple Lie group and so on?

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Since ACuriousMind has already answered some of the question before, I'll focus on how to compute or tabulated results.

https://groupprops.subwiki.org/wiki/Main_Page is available, but is a work in progress. Some of the groups, you might want are tabulated there.

SAGE has commands for calculating group cohomologies of reasonably small groups. Try small groups to test out how much computation power you can use. This works for point groups at least the ones I have tried running.

For crystallography groups see them tabulated in https://arxiv.org/abs/1612.00846

In groups that come in integer family like S_n, SU(n), SO(2n) you may see stabilization which makes the computation much easier. You need to have n big enough though.

AHusain
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