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Decomposition Numbers and Canonical Bases
Authors:Bernard Leclerc
Institution:(1) Département de Mathématiques, Université de Caen, 14032 Caen Cedex, France. e-mail
Abstract: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 
$$U_q \left( {\widehat{{\text{sl}}}_n } \right)$$
. It follows that these coefficients can all be expressed in terms of those of the global crystal basis of the irreducible subrepresentation generated by the vacuum vector. As a consequence, using the works of Ariki and Varagnolo and Vasserot, it is possible to give a new proof of Lusztig"s character formula for the simple U v (sl r )-modules at roots of unity, which does not involve representations of 
$$\widehat{{\text{sl}}}_r $$
of negative level.
Keywords:decomposition numbers  Schur algebras
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