The algebra of quasi-symmetric functions is free over the integers

Abstract

Let F denote the Leibniz-Hopf algebra, which also turns up as the Solomon descent algebra and the algebra of noncommutative symmetric functions. As an algebra F = Z〈Z1, Z2,...〉, the free associative algebra over the integers in countably many indeterminates. The coalgebra structure is given by μ(Zn) = Σni=0 Zi ⊗ Zn-i, Z0 = 1. Let M be the graded dual of F. This is the algebra of quasi-symmetric functions. The Ditters conjecture says that this algebra is a free commutative algebra over the integers. In this paper the Ditters conjecture is proved. © 2001 Elsevier Science.