Modular Lie powers and the Solomon descent algebra

الأصل · EN

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.