Quantcast
  • Register
PhysicsOverflow is a next-generation academic platform for physicists and astronomers, including a community peer review system and a postgraduate-level discussion forum analogous to MathOverflow.

Welcome to PhysicsOverflow! PhysicsOverflow is an open platform for community peer review and graduate-level Physics discussion.

Please help promote PhysicsOverflow ads elsewhere if you like it.

News

Migration to Bielefeld University was successful!

Please vote for this year's PhysicsOverflow ads!

Please do help out in categorising submissions. Submit a paper to PhysicsOverflow!

... see more

Tools for paper authors

Submit paper
Claim Paper Authorship

Tools for SE users

Search User
Reclaim SE Account
Request Account Merger
Nativise imported posts
Claim post (deleted users)
Import SE post

Users whose questions have been imported from Physics Stack Exchange, Theoretical Physics Stack Exchange, or any other Stack Exchange site are kindly requested to reclaim their account and not to register as a new user.

Public \(\beta\) tools

Report a bug with a feature
Request a new functionality
404 page design
Send feedback

Attributions

(propose a free ad)

Site Statistics

135 submissions , 113 unreviewed
3,803 questions , 1,343 unanswered
4,741 answers , 20,056 comments
1,470 users with positive rep
472 active unimported users
More ...

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

+ 5 like - 0 dislike
88 views

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 in Mathematics by fff123123 (25 points) [ no revision ]

1 Answer

+ 2 like - 0 dislike

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 by bolbteppa (110 points) [ revision history ]

Your answer

Please use answers only to (at least partly) answer questions. To comment, discuss, or ask for clarification, leave a comment instead.
To mask links under text, please type your text, highlight it, and click the "link" button. You can then enter your link URL.
Please consult the FAQ for as to how to format your post.
This is the answer box; if you want to write a comment instead, please use the 'add comment' button.
Live preview (may slow down editor)   Preview
Your name to display (optional):
Privacy: Your email address will only be used for sending these notifications.
Anti-spam verification:
If you are a human please identify the position of the character covered by the symbol $\varnothing$ in the following word:
p$\varnothing$ysicsOverflow
Then drag the red bullet below over the corresponding character of our banner. When you drop it there, the bullet changes to green (on slow internet connections after a few seconds).
To avoid this verification in future, please log in or register.




user contributions licensed under cc by-sa 3.0 with attribution required

Your rights
...