If it is possible to choose a particular representative \rho(g)\in\tilde\rho(g) in each family of operators in such a way that the homomorphism property is satisfied on the nose, rather than just up to a constant, then we say that \tilde\rho can be "de-projectivized", or that \tilde\rho can be "lifted to an ordinary representation". More concretely, we thus say that \tilde\rho can be de-projectivized if there are \rho(g)\in\tilde\rho(g) for each g\in G such that \rho(g)\rho(h)=\rho(gh). This possibility is discussed further below.
