Colored trees and noncommutative symmetric functions

TL;DR

This paper constructs algebra homomorphisms linking noncommutative symmetric functions, quasisymmetric functions, and the Ringel-Hall algebra of colored rooted forests, enriching the algebraic understanding of these combinatorial structures.

Contribution

It introduces new homomorphisms connecting noncommutative symmetric functions and quasisymmetric functions to the Ringel-Hall algebra of colored rooted forests, refining previous work by Zhao.

Findings

Established a homomorphism from a graded noncommutative symmetric functions algebra to the Ringel-Hall algebra.

Derived a dual homomorphism from the Connes-Kreimer Hopf algebra to quasisymmetric functions.

Provided a refinement of Zhao's earlier homomorphism in this algebraic context.

Abstract

Let $\CRF_S$ denote the category of $S$-colored rooted forests, and $\H_{\CRF_S}$ denote its Ringel-Hall algebra as introduced in \cite{KS}. We construct a homomorphism from a $K^+_0 (\CRF_S)$--graded version of the Hopf algebra of noncommutative symmetric functions to $\H_{\CRF_S}$. Dualizing, we obtain a homomorphism from the Connes-Kreimer Hopf algebra to a $K^+_0 (\CRF_S)$--graded version of the algebra of quasisymmetric functions. This homomorphism is a refinement of one considered by W. Zhao in \cite{Z}.