Modular Lie Powers and the Solomon descent algebra

Let V be an r-dimensional vector space over an infinite field F of prime characteristic p, and let Ln(V ) denote the n-th homogeneous component of the free Lie algebra on V . We study the structure of Ln(V ) as a module for the general linear group GLr(F ) when n = pk and k is not divisible by p and where n ≥ r. Our main result is an explicit 1-1 correspondence, multiplicity-preserving, between the indecomposable direct summands of Lk(V ) and the indecomposable direct summands of Ln(V ) which are not isomorphic to direct summands of V ⊗n. The direct summands of Lk(V ) have been parametrised earlier, by Donkin and Erdmann. Bryant and Stöhr have considered the case n = p but from a different perspective. Our approach uses idempotents of the Solomon descent algebras, and in addition a correspondence theorem for permutation modules of symmetric groups.