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arXiv · math/9902006

Decomposition numbers and canonical bases

Bernard Leclerc
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Abstract

We obtain some simple relations between decomposition numbers of quantized Schur algebras at an n-th 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 previously introduced in a joint work with Thibon. 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 U_v(sl_r)-modules at roots of unity, which does not involve representations of sl^_r of negative level.

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Bernard Leclerc. 1999-02-01. Decomposition numbers and canonical bases. https://arxiv.org/abs/math/9902006

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