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We show that the different labelings of the crystal graph for irreducible highest weight \(\mathcal{U}_q (\widehat{\mathfrak{sl}}_e)\)-modules lead to different labelings of the simple modules for non semisimple Ariki–Koike algebras by using Lusztig a-values.

Crystal Graphs of Higher Level q-deformed Fock Spaces, Lusztig a-values and Ariki–Koike Algebras