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Suppose $\rho : G \to PGL_n(k)$ is a projective representation of a group $G$ over a field $k$. It's classical that the obstruction to lifting this to a linear representation $G \to GL_n(k)$ is a cohomology class $[c] \in H^2(G, k^{\times})$.

Question: What's the name of this cohomology class?

It's tempting to call it the Schur class, but googling this name only produced a single paper from 1974 which used it in connection with projective representations. I haven't been able to find any other name in common use. If no one suggests a better name, that's the one I'll go with.

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    $\begingroup$ I think "the cohomology class associated with the projective representation" is a name that's in fairly common use. $\endgroup$
    – Jeremy Rickard
    Nov 12, 2015 at 9:08
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    $\begingroup$ So "Schur class" is much shorter... of course, it depends if you have to use it 3 times or 50 times in a single paper... $\endgroup$
    – YCor
    Nov 12, 2015 at 11:03
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    $\begingroup$ This kind of obstruction is called Mackey obstruction. By abuse of language we could call this cohomology class "the" Mackey obstruction. $\endgroup$
    – Paul Broussous
    Nov 12, 2015 at 11:59

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