On Young modules of general linear groups

We study ℓ-permutation modules of finite general linear groups GLn(q) acting on partial flags in the natural module, where the coefficient field of the modules has characteristic ℓ, for ℓ∤q. We call the indecomposable summands of these permutation modules linear Young modules. We determine their vertices and Green correspondents, by methods relying only on the representation theory of GLn(q).Furthermore, we show that when the multiplicative order of q modulo ℓ is strictly greater than 1, the Specht modules for GLn(q) in characteristic ℓ form a stratifying system. This implies in particular, that for GLn(q)-modules with Specht filtration, the filtration multiplicities are independent of the filtration. This is an analogue of a recent theorem by Hemmer and Nakano.