Qsym over Sym is free

We study here the ring QSn of Quasi-symmetric functions in the variables x1,x2,…,xn. Bergeron and Reutenauer (personal communication) formulated a number of conjectures about this ring; in particular, they conjectured that it is free over the ring Λn of symmetric functions in x1,x2,…,xn. We present here an algorithm that recursively constructs a Λn-module basis for QSn thereby proving one of the Bergeron–Reutenauer conjectures. This result also implies that the quotient of QSn by the ideal generated by the elementary symmetric functions has dimension n!. Surprisingly, to show the validity of our algorithm we were led to a truly remarkable connection between QSn and the harmonics of Sn.