There are several theories for the fractional quantum Hall effect. The last listed in the Wikipedia article are composite fermions, though these seem to be a subset of fractional exchange statistics:
A finite group of anyons has some character that defines their Cayley graph. But you can view $n$ vortices as a rose, whose universal cover is a Cayley graph. The former is more general.
What about a hierarchy of states? You can generalize the idea of a boundary in algebraic toplogy to use some character, similar to the immanant (also see here). I want to say that every local conformation can be described by an immanant, i.e. character, but I'm not sure.
