A decomposition of Solomon's descent algebra

Abstract

A descent class, in the symmetric group S n , is the collection of permutations with a given descent set. It was shown by L. Solomon ( J. Algebra 41 (1976), 255–264) that the product (in the group algebra Q( S n )) of two descent classes is a linear combination of descent classes. Thus descent classes generate a subalgebra of Q( S n ). We refer to it here as Solomon's descent algebra and denote it by Σ n . This algebra is not semisimple but it has a faithul representation in terms of upper triangular matrices. The main goal of this paper is a decomposition of its multiplicative structure. It develops that Σ n acts in a natural way on the so-called Lie monomials. This action has a purely combinatorial description and is a crucial tool in the construction of a complete set of indecomposable representations of Σ n . In particular we obtain a natural basis of irreducible orthogonal idempotents Σ λ (indexed by partitions of n) for the quotient Σ n √Σ n . Natural bases of nilpotents and idempotents for the subspaces E λ Σ n E μ , for two arbitrary partitions λ and μ, are also constructed and the dimensions of these spaces are given a combinatorial interpretation in terms of the so-called decreasing factorization of an arbitary word into a product of Lyndon words.