# Criterion for projective representation to lift to linear representation

## 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

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

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

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

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