 
Summary: STRUCTURE OF THE MALVENUTOREUTENAUER
HOPF ALGEBRA OF PERMUTATIONS
MARCELO AGUIAR AND FRANK SOTTILE
Abstract. We analyze the structure of the MalvenutoReutenauer Hopf algebra SSym
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 M¨obius inversion on the weak order on the symmetric
groups. This is in analogy with the relationship between the monomial and funda
mental 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.
Introduction 1
1. Basic definitions and results 3
