The module structure of Solomon's descent algebra

Abstract

A close connection is uncovered between the lower central series of the free associative algebra of countable rank and the descending Loewy series of the direct sum of all Solomon descent algebras Δn, n ∈ ℕ0, Each irreducible Δn-module is shown to occur in at most one Loewy section of any principal indecomposable Δn-module. A precise condition for this occurrence and formulae for the Cartan numbers are obtained.