Decomposition Numbers and Canonical Bases

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• 作者：Leclerc ; Bernard
• 刊名：Algebras and Representation
• 出版年：2000
• 出版时间：2000
• 年：2000
• 卷：3
• 期：3
• 页码：277-287
• 全文大小：90 KB

文摘

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 $U\_\{q\}(\backslash widehat\{\backslash mathrm\{sl\}\}\_\{n\})$. 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 $\backslash widehat\{\backslash mathrm\{sl\}\}\_\{r\}$ of negative level.