Structure of the Malvenuto-reutenauer Hopf Algebra of Permutations

We analyze the structure of the Malvenuto-Reutenauer Hopf algebraSSym 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. Introduction 1 1. Basic definitions and results 3 2. The weak order on the symmetric group 7 3. The coproduct of SSym 14 4. The product of SSym 15 5. The antipode of SSym 18 6. Cofreeness, primitive elements, and the coradical filtration of SSym 23 7. The descent map to quasi-symmetric functions 27 8. SSym is a crossed product over QSym 31 9. Self-duality of SSym and applications 34 References 39 Introduction Malvenuto [22] introduced the Hopf algebra SSym of permutations, which has a linear basis {Fu | u ∈ Sn, n ≥ 0} indexed by permutations in all symmetric groups Sn. The Hopf algebra SSym is non-commutative, non-cocommutative, self-dual, and graded. Among its sub-, quotient-, and subquotientHopf algebras are many algebras central to algebraic combinatorics. These include the algebra of symmetric functions [21, 33], Gessel’s algebra QSym of quasi-symmetric functions [13], the algebra of non-commutative symmetric functions [12], the Loday-Ronco algebra of planar binary trees [19], Stembridge’s algebra of peaks [34], the Billera-Liu algebra of Eulerian enumeration [2], and 2000 Mathematics Subject Classification. Primary 05E05, 06A11, 16W30; Secondary 05E15, 06A07, 06A15.