(1) Department of Mathematics, Texas A&M University, College Station, TX 77843, USA
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.
Aguiar supported in part by NSF grant DMS-0302423. Sottile supported in part by NSF CAREER grant DMS-0134860, the Clay Mathematics
Institute, and MSRI.