A projective representation of a group $G$ is a group homomorphism $\theta:G \to \text{PGL}(V)$. I am trying to show that there's a bijection between the set of all projective representations and $H^2(G,\mathbb{C}^{\times})$. For the first direction, this goes as follows:
Let ProjRep denote the set of all projective representations of a group $G$. Let $\theta$ be a projective representation. Then, a lift $T$ of $\theta$ exists, i.e. there exists a map $T:G \to \text{GL}(V)$, such that $T(g) T(h)=\alpha(g,h)T(gh)$, where $\alpha$ is a 2-cocycle. Note that the data of $\alpha$ comes for free so now we can denote the lift as $(T,\alpha)$. Moreover, if $(L,\beta)$ is another lift, then $L(g)$ and $T(g)$ are scalar multiples of each other and $\alpha$ and $\beta$ are cohomologous. To this end, we can define a map:
\begin{equation}
M:\text{ProjRep} \to H^2(G,\mathbb{C}^{\times}), \; \; M(\theta):=[\alpha],
\end{equation}
where $\alpha$ is the cocycle associated to any lift of $\theta$.
The question then reduces to the following: given a cohomology class $[\alpha] \in H^2(G,\mathbb{C}^{\times})$, does there exist a projective representation, to which we can associate this cocycle? Does every cocycle come from a projective representation?
Best Answer
A lift $T:G\rightarrow GL(V)$ of $\theta$ is any map such that $[T(g)]=\theta(g)$, where $[A]$ denotes the equivalence class of $GL(V)$ under $A\sim B$ if $A=\lambda B$ for $\lambda \in \mathbb{C}^*$. If $T_1$ and $T_2$ are two lifts, it follows that for any $g\in G$ there exists $\lambda(g)\in \mathbb{C}^*$ such that $T_1(g)=\lambda(g)T_2(g)$. This defines a function $\lambda : G\rightarrow \mathbb{C}^*$, namely a 1-cochain. If $T_1$ is characterize by the product law $$ T_1(gh)=\alpha _1(g,h)T_1(g)T_1(h)$$ with $\alpha _1$ a 2-cocycle, then you have $$ T_2(g,h)=\frac{\lambda(g)\lambda(h)}{\lambda(gh)}\alpha_1(g,h)T_2(g)T_2(h)=\alpha_2(g,h)T_2(g)T_2(h) $$ where $\alpha _2=d\lambda \cdot \alpha _1$, namely $[\alpha _1]=[\alpha_2]\in H^2(BG,\mathbb{C}^*)$. Thus all possible lifts of $\theta$ are characterize by all possible projective phases in a given cohomology class. The correct statement is that if the cohomology class vanishes, then you can choose uniquely a lift which is a linear representation. This is a choice: not all lifts will be linear representations. Moreover any two lifts which are linear representation will be related as $T_1(g)=\lambda(g)T_2(g)$, with $\lambda \in H^1(BG,\mathbb{C}^*)$, namely a group homomorphsim $\lambda :G \rightarrow \mathbb{C}^*$.