Composition factors of Specht modules for Hecke algebras of type An
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- 作者：Eugene Murphy
- 关键词：Hecke algebras ; Specht modules ; Composition factors
- 刊名：Journal of Algebra
- 出版年：2006
- 出版时间：1 December 2006
- 年：2006
- 卷：306
- 期：1
- 页码：268-289
- 全文大小：249 K
文摘
We examine the composition factors of Specht modules for Hecke algebras of type An at roots of zero, and their positions in the Jantzen–Schaper filtration. Each Specht module is decomposed into a direct sum of orthogonal subspaces corresponding to residue classes of standard tableaux; similarly for the Gram matrix. We show that, for a given subset of these classes, the corresponding invariant factors of the Gram matrix over a local ring completely determine the decomposition matrix and Jantzen–Schaper filtration.
From this we deduce elementary proofs for a number of well-known results, notably the James “first column” theorem together with its generalisation by Donkin, and the determination of the decomposition matrices for two-part partitions for the symmetric group algebra and for general linear groups. We extend these to analogous results for the Jantzen–Schaper filtration; in particular, we derive a closed formula for the Jantzen indices for two-part partititions (two-column diagrams in our formulation) in the case of the Hecke algebra.