  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. In addition, we describe the structure
constants of the multiplication as a certain number of facets of the
permutahedron. As a consequence we obtain a new interpretation of the product
of monomial quasi-symmetric functions in terms of the facial structure of the
cube. The Hopf algebra of Malvenuto and Reutenauer has a linear basis indexed
by permutations. Our results are obtained from a combinatorial description of
the Hopf algebraic structure with respect to a new basis for this algebra,
related to the original one via M\"obius inversion on the weak order on the
symmetric groups. This is in analogy with the relationship between the monomial
and fundamental bases of the algebra of quasi-symmetric functions. Our results
reveal a close relationship between the structure of the Malvenuto-Reutenauer
Hopf algebra and the weak order on the symmetric groups.
