A Noncommutative Symmetric System over the Grossman-Larson Hopf Algebra of Labeled Rooted Trees

Abstract

In this paper, we construct explicitly a noncommutative symmetric ($\mathcal{N}$CS) system over the Grossman-Larson Hopf algebra of labeled rooted trees. By the universal property of the $\mathcal{N}$CS system formed by the generating functions of certain noncommutative symmetric functions, we obtain a specialization of noncommutative symmetric functions by labeled rooted trees. Taking the graded duals, we also get a graded Hopf algebra homomorphism from the Connes-Kreimer Hopf algebra of labeled rooted forests to the Hopf algebra of quasi-symmetric functions. A connection of the coefficients of the third generating function of the constructed $\mathcal{N}$CS system with the order polynomials of rooted trees is also given and proved.

Latex, 30 pages. Following the referees' suggestions, several places have been improved. In particular, some diagrams of rooted trees have been added. To appear in J. Alg. Comb

References in corpus (5)

• Combinatorial Hopf algebras and generalized Dehn-Sommerville relations
• Combinatorics of Rooted Trees and Hopf Algebras
• NCS Systems over Differential Operator Algebras and the Grossman-Larson Hopf Algebras of Labeled Rooted Trees
• Noncommutative Symmetric Functions and the Inversion Problem
• A New Approach to Order Polynomials of Labeled Posets and Their Generalizations

Cited by in corpus (6)

• Two interacting Hopf algebras of trees
• The Magnus expansion, trees and Knuth's rotation correspondence
• On an extension of Knuth's rotation correspondence to reduced planar trees
• Hopf algebra structure of generalized quasi-symmetric functions in partially commutative variables
• Colored trees and noncommutative symmetric functions
• On the sub-adjacent Hopf algebra of the universal enveloping algebra of a post-Lie algebra