Abstract :
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.
