  We make a systematic study of filtrations of a free group F defined as
products of powers of the lower central series of F. Under some assumptions on
the exponents, we characterize these filtrations in terms of the group algebra,
the Magnus algebra of non-commutative power series, and linear representations
by upper-triangular unipotent matrices. These characterizations generalize
classical results of Grun, Magnus, Witt, and Zassenhaus from the 1930's, as
well as later results on the lower p-central filtration and the p-Zassenhaus
filtrations. We derive alternative recursive definitions of such filtrations,
extending results of Lazard. Finally, we relate these filtrations to Massey
products in group cohomology.
