# Reference recommendation for Projective representation, group cohomology, Schur's multiplier and central extension

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?

• This answer of mine answers all these questions except for "how to compute group cohomology" (and it doesn't really call $H^2$ "group cohomology", but that's what it is). May 17, 2017 at 9:02
• Having book-size sources for this kind of material would still be quite valuable @ACuriousMind
– Danu
May 17, 2017 at 12:42
• @Danu I said nothing to the contrary; I just wanted to point out that the "non-resource recommendation" version of this question basically already exists on the site, at least for some of these questions. May 17, 2017 at 13:17
• @ACuriousMind Thank you. Do you know which book can discuss the theory of projective representation thoroughly and give many examples like many textbooks of group representation for physicist.
– user153663
May 18, 2017 at 16:59

## 1 Answer

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.