Decomposition numbers and canonical bases

We obtain some simple relations between decomposition numbers of quantized Schur algebras at an nth root of unity (over a field of characteristic 0). These relations imply that every decomposition number for such an algebra occurs as a decomposition number for some Hecke algebra of type A. We prove similar relations between coefficients of the canonical basis of the q-deformed Fock space representation of Uq(ŝln) introduced in [20]. It follows that these coefficients can all be expressed in terms of those of the global crystal basis of the irreducible sub-representation generated by the vacuum vector. As a consequence, using works of Ariki and Varagnolo-Vasserot, it is possible to give a new proof of Lusztig’s character formula for the simple Uv(slr)-modules at roots of unity, which does not involve representations of ŝlr of negative level.