The Algebra of Quasi-Symmetric Functions Is Free over the Integers

Abstract

Let Z denote the Leibniz–Hopf algebra, which also turns up as the Solomon descent algebra and the algebra of noncommutative symmetric functions. As an algebra Z=Z〈Z1, Z2,…〉, the free associative algebra over the integers in countably many indeterminates. The coalgebra structure is given by μ(Zn)=∑ni=0Zi⊗Zn−i, Z0=1. Let M be the graded dual of Z. 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.