The Solomon descent algebra

Abstract

Abstract The convolution subalgebra generated by the projections of the free associative algebra arising from its graduation is also an algebra under composition, and is canonically isomorphic with a subalgebra of the symmetric group algebra, called the descent algebra; the latter was introduced by Solomon for each finite Coxeter group. The canonical projections of the free associative algebra, arising from its structure of enveloping algebra of the free Lie algebra, correspond to the primitive idempotents of the descent algebra; this is established in Section 9.2, where some insight into the structure of this algebra is also given. In the next section, we study various homomorphisms of this algebra: one is a mapping into the ring of symmetric functions, and its kernel is the radical of the descent algebra; another maps each descent algebra into another one, and is a derivation with respect to the convolution product. Elements of the descent algebra are also characterized, in the symmetric group algebra, by their action on Lie monomials. In the final section, we introduce quasisymmetric functions, which are closely related to the descent algebra, and we give an application to the enumeration of permutations.