Structure of the MalvenutoReutenauer Hopf algebra of permutations
Abstract
We analyze the structure of the MalvenutoReutenauer 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 quasisymmetric 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 quasisymmetric 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 quasisymmetric functions. Our results reveal a close relationship between the structure of the MalvenutoReutenauer Hopf algebra and the weak order on the symmetric groups.
 Publication:

arXiv Mathematics eprints
 Pub Date:
 March 2002
 DOI:
 10.48550/arXiv.math/0203282
 arXiv:
 arXiv:math/0203282
 Bibcode:
 2002math......3282A
 Keywords:

 Mathematics  Combinatorics;
 Mathematics  Quantum Algebra;
 Mathematics  Rings and Algebras;
 05E05;
 06A11;
 16W30
 EPrint:
 40 pages, 6 .eps figures. Full version of math.CO/0203101. Error in statement of Lemma 2.17 in published version corrected