  We examine the crystal graph of the $\widehat{\mathfrak{sl}}_e$-module
arising from an $\widehat{\mathfrak{sl}}_e$-categorification to study the
defining endo-functors of the categorification. This yields lower bounds on the
number of irreducible constituents of certain objects. We use Ariki's
categorification result on Ariki-Koike algebras to obtain a new lower bound on
the number of constituents of their parabolically induced modules. In
particular this will imply reducibility of every induced module.
