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

TL;DR

This paper explores the relationships between various Hopf algebras related to rooted trees and symmetric functions, introducing homomorphisms that connect these algebraic structures in a unified framework.

Contribution

It constructs a commutative diagram linking the Connes-Kreimer, Grossman-Larson, and noncommutative Hopf algebras with symmetric functions, revealing new structural insights.

Findings

Established a homomorphism linking rooted tree Hopf algebras to symmetric functions.

Demonstrated the self-duality of Foissy's noncommutative Hopf algebra.

Connected algebraic structures via a commutative diagram.

Abstract

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