Branching properties for the groups G(de,e,r)

We study general properties of the restriction of the representations of the finite complex reflection groups G(de,e,r+1) to their maximal parabolic subgroups of type G(de,e,r), and focus notably on the multiplicity of components. In combinatorial terms, this amounts to the following question: which symmetries arise or disappear when one changes (exactly) one pearl in a combinatorial necklace?