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Recent work on perturbative quantum field theory has led to much study of theConnes-Kreimer Hopf algebra. Its (graded) dual, the Grossman-Larson Hopfalgebra of rooted trees, had already been studied by algebraists. L. Foissyintroduced a noncommutative version of the Connes-Kreimer Hopf algebra, whichturns out to be self-dual. Using some homomorphisms defined by the author andW. Zhao, we describe a commutative diagram that relates the aforementioned Hopfalgebras to each other and to the Hopf algebras of symmetric functions,noncommutative symmetric functions, and quasi-symmetric functions.

Rooted trees and symmetric functions: Zhao's homomorphism and the commutative hexagon