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
liniear 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
Möbius 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.