Structure of the Malvenuto-Reutenauer Hopf algebra of permutations (Extended Abstract)

Abstract

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.

12 pages, 2 .eps figures. (minor revisions) Extended abstract for Formal Power Series and Algebraic Combinatorics, Melbourne, July 2002