On 2-modular representations of the symmetric groups

Let G=GLn denote the general linear group of invertible n×n matrices with entries in an algebraically closed field F of characteristic 2. We prove the existence of certain composition factors in the restriction to GLn−k×GLk of some polynomial G-modules. As a consequence, we resolve the final case of the problem, addressed by Jantzen and Seitz, of when an irreducible modular representation of the symmetric group Sn remains irreducible upon restriction to the Young subgroup Sn−k×Sk.