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

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

2002-03-11

arxiv.org/abs/math/0203101v2

Mathematics

Combinatorics

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

Scientific paper

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.

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