  We analyze the structure of the Malvenuto-Reutenauer Hopf algebra of
permutations in detail. We give explicit formulas for its antipode, prove that
it is a cofree coalgebra, determine its primitive elements and its coradical
filtration and show that it decomposes as a crossed product over the Hopf
algebra of quasi-symmetric functions. We also describe the structure constants
of the multiplication as a certain number of facets of the permutahedron. Our
results reveal a close relationship between the structure of this Hopf algebra
and the weak order on the symmetric groups.
