We show that the canonical basis associated to any highest weight
U_{v}(hat{sl}_{e})-module can be decomposed on the canonical basis of its
corresponding U_{v}({sl}_{\infty})-module. We establish that the transition
matrix associated to this decomposition is unitriangular with coefficients in
Z[v] and give a procedure to compute them. We conjecture these coefficients are
in fact in N[v]. This provides a natural quantization of a theorem by Geck and
Rouquier on the factorization of decomposition matrices associated to
Ariki-Koike algebras.