Let $G$ be a finite group and let $D(g)$ be a projective representation of $G$ i.e.
\begin{equation}
D(g) D(h) = e^{i \omega(g,h)} D(gh)
\end{equation}
These can be classified by the equivalence relation $\omega(g,h) \sim \omega(g,h)+\theta(g)+ \theta(h) - \theta(gh)$ subject to the condition $\omega(g,h)+\omega(gh,l)-\omega(h,l)-\omega(g,hl) = 0$.
The distinct equivalent classes of projective representations are labeled by elements of $H^2(G,U(1))$.
I know that for every such finite group, there is atleast one finite covering group C with the property that every projective representation of G can be lifted to an ordinary representation of C [1]. 


My question is about the relationship between the irreducible representations (irreps) of C and the group $H^2(G,U(1))$. Specifically, is the following statement true?:


Every irrep $\Gamma\_i$ of C can be associated an element of $\nu \in H^2(G,U(1))$ like $\Gamma^\nu\_i$. The group property of $H^2(G,U(1))$ is reflected in the Clebsch-Gordan decomposition of tensor product of irreps of C:
\begin{equation}
\Gamma^\nu\_i \otimes \Gamma^\mu\_j \cong \bigoplus\_k \Gamma^{\nu+\mu}\_k
\end{equation}


I have noticed that this is true for all cases I have seen when $ H^2(G,U(1)) \cong \mathbb{Z}\_2$ like $G = \mathbb{Z}\_2 \times \mathbb{Z}\_2$, $C = D\_8$ but I am unsure if this is true in general. 


[1] <https://en.wikipedia.org/wiki/Schur_multiplier#Relation_to_projective_representations>


PS: I am a physicist and my interest in the above question comes from its use in condensed matter physics. 



Relationship between irreducible representations of the Schur covering group and elements of $H^2(G,U(1))$
The answer to your question is **Yes.** Consider your covering group $C$ as a central extension:
$$1 \to N \to C \to G \to 1$$
and suppose it is given by a 2-cocycle $\alpha \in H^2(G, N)$. Then for any homomorphism $f: N \to \mathbb{C}^\times$, we get a 2-cocycle $f \circ \alpha \in H^2(G,\mathbb{C}^\times)$ by composition of maps, and this yields a homomorphism
$$\operatorname{Hom}(N,\mathbb{C}^\times) \to H^2(G,\mathbb{C}^\times).$$
You are assuming that $C$ is a "representation group", which amounts to the property that this homomorphism is an isomorphism. Thus, any projective representation of $G$ lands in a commutative diagram
$$\require{AMScd}
\begin{CD}
1 @>>> N @>>> C @>>> G @>>> 1 \\
@. @VVV @VVV @VVV \\
1 @>>> \mathbb{C}^\times @>>> GL(V) @>>> PGL(V) @>>> 1
\end{CD}$$
The 2-cocycle for your projective representation corresponds to the 1-dimensional representation of $N$ given by the leftmost vertical arrow. Any tensor product of representations of $C$ induces a tensor product of 1-dimensional representations of $N$. Tensor product of 1-dimensional representations of $N$ yields the group law on $\operatorname{Hom}(N,\mathbb{C}^\times)$, and by our assumption, this is the group law on $H^2(G, \mathbb{C}^\times)$.


I found all the statements I needed in [this exposition by Mendonca](http://alistairsavage.ca/pubs/Mendonca-Projective_Representation_of_Groups.pdf).


