Criterion for projective representation to lift to linear representation
Contents
Statement
Suppose is a finite group and is a central subgroup of . Suppose and is a homomorphism of groups, and hence a projective representation of . Let and be the obvious quotient maps. We call a linear representation a lift of if .
The statement is that there exists a linear representation that is a lift of if and only if the following holds.
Description of condition in cohomology language
Consider the following biadditive map:
Fixing the element of corresponding to the extension group , we get a homomorphism:
from the group of one-dimensional representations of the central subgroup to .
The projective representation lifts to a linear representation if and only if the image of the above homomorphism contains the cohomology class (i.e., the element of corresponding to .
Description of condition in homology language
Consider the formula for second cohomology group for trivial group action in terms of Schur multiplier and abelianization:
The extension group corresponds to an element of , which hence maps to an element of . By composition, this defines a map:
The condition is that the image of this homomorphism should contain the cohomology class corresponding to the projective representation.
Particular cases
Case | Conclusion |
---|---|
the central subgroup is trivial. | In this case, a projective representation lifts to a linear representation if and only if its cohomology class is trivia. |
is a Schur covering group of . | In this case, every projective representation of lifts to a linear representation of . |
References
- Character Theory of Finite Groups by I. Martin Isaacs, ISBN 0486680142, Page 182, Theorem 11.13, ^{More info}