TITLE: How do small central extensions drop the dimension of a faithful representation?
QUESTION [9 upvotes]: Apologies in advance that this is a very soft question. I might be talking complete nonsense. But I hope I am talking about something that has even been studied...
I am interested in the phenomenon in representation theory that a group whose faithful representations (over $\mathbb{C}$, say) have minimum dimension $n$ often has a central extension with faithful representations in dimensions smaller than $n$. For example, $S_4$ has no faithful representation in dimension less than 3, but it has a double cover $\tilde S_4$ with a faithful 2-dimensional representation. This phenomenon is striking to me because it takes "more information" to write down the elements of the bigger group, since an element of $\tilde A_5$ specifies an element of $A_5$, but not vice versa. So the dimension of the representation does not correspond to the amount of information it contains.
One might speculate that the "extra information" in the smaller-dimensional representation of the bigger group is housed in the (smallest) field over which the representation is defined; in my example, the faithful three-dimensional representations of $S_4$ are defined over $\mathbb{Q}$, but the faithful two-dimensional representations of $\tilde S_4$ I believe require $\mathbb{Q}(i)$.
But I'm unsatisfied by this explanation. First of all, although I do not have an example on hand, I am told (by Derek Holt) that one observes the same phenomenon with permutation representations, with degree replacing dimension, where there is no question of the field of definition. Secondly, it begs the question (to me anyway) of "how" the central extension "causes" the information to be moved from the size of the space to the size of the field.
My question is this:

Has this phenomenon attracted any research attention? Can you point me to any exploration of it in the literature?

Thanks in advance.

REPLY [6 votes]: I think the basic reason that this happens is quite simple. Matrix groups, such as ${\rm GL}(n,{\mathbb C})$ have a centre, consisting of the scalar matrices and so it natural for groups with nontrivial centres to have representations in which their centre maps to the scalars.
More to the point, the perfect group ${\rm SL}(n,{\mathbb C})$ has a cyclic centre of order $n$. So finite groups $G$ with cyclic subgroups $Z < Z(G) \cap [G,G]$ of order dividing $n$ are good candidates for mapping into ${\rm SL}(n,{\mathbb C})$.
So, for example, ${\rm SL}(2,5)$ with centre of order $2$ maps into ${\rm SL}(2,{\mathbb C})$, whereas the smallest dimension of a faithrful representation of $A_5={\rm PSL}(2,5)$ is $3$.
Or $3.A_6 < {\rm SL}(3,{\mathbb C})$ and $2.A_6 = {\rm SL}(2,9) < {\rm SL}(4,{\mathbb C})$, but the smallest degree nontrivial representation of $A_6$ is $5$.