Cocommutative Hopf Algebras of Permutations and Trees

Authors:
Marcelo Aguiar;Frank Sottile
Affiliations:
Department of Mathematics, Texas A&M University, College Station, USA 77843;Department of Mathematics, Texas A&M University, College Station, USA 77843
Venue:
Journal of Algebraic Combinatorics: An International Journal
Year:
2005

Citing 3
Cited 1

Enumerative combinatorics

Enumerative combinatorics
Quasi-Shuffle Products

Journal of Algebraic Combinatorics: An International Journal
Order Structure on the Algebra of Permutations and of Planar Binary Trees

Journal of Algebraic Combinatorics: An International Journal

Commutative combinatorial Hopf algebras

Journal of Algebraic Combinatorics: An International Journal

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Abstract

Consider the coradical filtrations of the Hopf algebras of planar binary trees of Loday and Ronco and of permutations of Malvenuto and Reutenauer. We give explicit isomorphisms showing that the associated graded Hopf algebras are dual to the cocommutative Hopf algebras introduced in the late 1980's by Grossman and Larson. These Hopf algebras are constructed from ordered trees and heap-ordered trees, respectively. These results follow from the fact that whenever one starts from a Hopf algebra that is a cofree graded coalgebra, the associated graded Hopf algebra is a shuffle Hopf algebra.