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

+ 6 like - 0 dislike

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? 

asked May 18, 2017 in Mathematics by fff123123 (30 points) [ no revision ]

1 Answer

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To quote the relevant bits of Butler, Point Group Symmetry Applications: Methods and Tables, Sec. 2.6:

Around the turn of the century Schur gave a method for finding representations of a finite group in terms of fractional linear transformations (projective transformations). These retain the group multiplication law, but the space on which they act is a projective space (as in projective geometry), not a linear vector space..

Hamermesh (1962, Chapter 12) shows that fractional linear representations are equivalent to projective representations as usually defined, e.g., in space group theory.... This in turn is equivalent to having a set of p matrices, scalar multiples of each other, representing each group element. This is a p-valued representation...

...Cartan in 1913 developed a general method of constructing projective matrix irreps for the continuous groups. The continuity condition may be used to show that only the group of orthogonal transformations in n dimensions has nontrivial projective representations, ± 1 only, so the representations are at most double-valued (Littlewood 1950, p. 248).

Then quoting the beginning of Hamermesh, Group Theory and its Application to Physical Problems, Ch. 12:

It is remarkable that the problem of finding the ray representations of finite groups was stated and completely solved long before the advent of quantum mechanics. In a series of papers, Schur gave the general method for finding the irreducible representations of a finite group in terms of fractional linear transformations (projective transformations, collineations).

and it merges nicely with Weinberg's exposition.

answered Jun 13, 2017 by bolbteppa (120 points) [ revision history ]

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