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Structure of the Malvenuto-Reutenauer Hopf algebra of permutations. (English) Zbl 1056.05139
Summary: We analyze the structure of the Malvenuto-Reutenauer Hopf algebra $${\mathfrak S}$$Sym 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 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.

##### MSC:
 05E05 Symmetric functions and generalizations 06A11 Algebraic aspects of posets 05E15 Combinatorial aspects of groups and algebras (MSC2010) 06A07 Combinatorics of partially ordered sets 06A15 Galois correspondences, closure operators (in relation to ordered sets) 16W30 Hopf algebras (associative rings and algebras) (MSC2000)
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