Free polynomial generators for the Hopf algebra \(QSym\) of quasisymmetric functions (Q1963995)

A word \(w\) in letters from \(\mathbb N_+\), the set of positive integers, is a Lyndon word if for any non-trivial factorization \(w=xy\), one has \(y>w\) in the lexicographical order on words. Let \(Q\text{Sym}\) be the ring of quasisymmetric functions over the rational numbers \(Q\). \textit{C. Malvenuto} and \textit{C. Reutenauer} showed that \(Q\text{Sym}\) is the free commutative polynomial algebra on the algebraically independent set of Lyndon words [J. Algebra 177, No.~3, 967--982 (1995; Zbl 0838.05100)]. See also \textit{C. Reutenauer}'s book ``Free Lie algebras'' [Oxford: Clarendon Press (1993; Zbl 0798.17001)] for this result (Corollary 2) as well as for a discussion of \(Q\text{Sym}\) (Section 9.4). In the paper under review, the authors give a definition of a modified Lyndon word, and then show that as a commutative algebra, \(Q\text{Sym}\) is freely generated over \(\mathbb Z\), the integers, by the algebraically independent set of modified Lyndon words. As a corollary, \(Q\text{Sym}\) is a free module over \(\text{Sym}\), the algebra of symmetric functions.