Criterion for projective representation to lift to linear representation

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Statement

Suppose G is a finite group and N is a central subgroup of G. Suppose Q = G/N and \rho: Q \to PGL_d(\mathbb{C}) is a homomorphism of groups, and hence a projective representation of G/N. Let \alpha:G \to Q and \pi_d:GL_d(\mathbb{C}) \to PGL_d(\mathbb{C}) be the obvious quotient maps. We call a linear representation \theta: G \to GL_d(\mathbb{C}) a lift of \rho if \pi_d \circ \theta = \rho \circ \alpha.

The statement is that there exists a linear representation that is a lift of \rho if and only if the following holds.

Description of condition in cohomology language

Consider the following biadditive map:

H^2(Q;N) \times \operatorname{Hom}(N,\mathbb{C}^*) \to H^2(Q;\mathbb{C}^*)

Fixing the element of H^2(Q;N) corresponding to the extension group G, we get a homomorphism:

\operatorname{Hom}(N,\mathbb{C}^*) \to H^2(Q;\mathbb{C}^*)

from the group of one-dimensional representations of the central subgroup N to H^2(Q;\mathbb{C}^*).

The projective representation \rho lifts to a linear representation if and only if the image of the above homomorphism contains the cohomology class (i.e., the element of H^2(Q;\mathbb{C}^*) corresponding to \rho.

Description of condition in homology language

Consider the formula for second cohomology group for trivial group action in terms of Schur multiplier and abelianization:

0 \to \operatorname{Ext}^1(Q^{\operatorname{ab}},N) \to H^2(Q;N) \to \operatorname{Hom}(H_2(Q;\mathbb{Z}),N) \to 0

The extension group G corresponds to an element of H^2(Q;N), which hence maps to an element of \operatorname{Hom}(H_2(Q;\mathbb{Z}),N). By composition, this defines a map:

\operatorname{Hom}(N,\mathbb{C}^*) \to \operatorname{Hom}(H_2(Q;\mathbb{Z}),\mathbb{C}^*) = H^2(Q;\mathbb{C}^*)

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.
G is a Schur covering group of Q. In this case, every projective representation of Q lifts to a linear representation of G.

References

  • Character Theory of Finite Groups by I. Martin Isaacs, ISBN 0486680142, Page 182, Theorem 11.13, More info