Embeddings of higher Lie modules

The higher Lie modules of the general linear group GL(V) over a finite dimensional vector space V arise naturally from the Poincaré–Birkhoff–Witt basis of the tensor algebra over V. They are indexed by partitions. For the higher Lie modules corresponding to hook partitions of n a complete chain of embeddings is obtained. As an application, a new inductive proof of Klyachkoʹs result on the irreducible components of the classical Lie module is given. Additionally, all irreducible components of multiplicity 1 are determined.