Modular Lie powers and the Solomon descent algebra

摘要

Let V be an r-dimensional vector space over an infinite field F of prime characteristic p, and let L n (V) denote the nth homogeneous component of the free Lie algebra on V. We study the structure of L n (V) as a module for the general linear group GL r (F) when n=pk and k is not divisible by p and where r≥n. Our main result is an explicit 1-1 correspondence, multiplicity-preserving, between the indecomposable direct summands of L k (V) and the indecomposable direct summands of L n (V) which are not isomorphic to direct summands of V ⊗ n . Our approach uses idempotents of the Solomon descent algebras, and in addition a correspondence theorem for permutation modules of symmetric groups.