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STRUCTURE OF THE MALVENUTO-REUTENAUER HOPF ALGEBRA OF PERMUTATIONS
 

Summary: STRUCTURE OF THE MALVENUTO-REUTENAUER
HOPF ALGEBRA OF PERMUTATIONS
MARCELO AGUIAR AND FRANK SOTTILE
Abstract. We analyze the structure of the Malvenuto-Reutenauer 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 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 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 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

  

Source: Aguiar, Marcelo - Department of Mathematics, Texas A&M University

 

Collections: Mathematics