In general, if I understand correctly, the representation theory of the braid groups is quite complicated, and there's no classification of the irreducibles. However, the braid groups form a sort of system of groups just as the symmetric groups do, and so one can ask about representation stability for coherent systems of representations of braid groups. One example might be the sequence of homologies of configuration spaces. Is anything at all known about when, if ever, representation stability occurs in this setting? For instance, is it known perhaps that the sequence of homologies of configuration spaces cannot exhibit representation stability? This is a very basic question, mostly just a reference request.

Note that I am not asking about representation stability for the braid groups themselves, which is known, nor am I asking about representation stability for the sequence of homologies of configuration spaces as symmetric group representations, which also is known.

In general, if I understand correctly, the representation theory of the braid groups is quite complicated, and there's no classification of the irreducibles. However, the braid groups form a sort of system of groups just as the symmetric groups do, and so one can ask about representation stability for coherent systems of representations of braid groups. Editing in response to Andy's comments below: it may be that "representation stability" doesn't mean anything because we can't decompose braid group representations in the same way we can decompose symmetric group representations. So a more basic question would be: is there any sensible way to talk about systems of braid group representations stabilizing that doesn't involve decomposition into irreducibles? 

This is a very basic question, mostly just a reference request.

Note that I am not asking about representation stability for the braid groups themselves.