On weak peak quasisymmetric functions

TL;DR

This paper introduces weak peak quasisymmetric functions within the framework of weak composition quasisymmetric functions, establishing their algebraic structures and relationships, including bases, Hopf algebra properties, and Rota-Baxter algebra embeddings.

Contribution

It constructs the weak peak quasisymmetric functions, identifies their algebraic structures, and explores their embeddings and quotients within the Hopf and Rota-Baxter algebra frameworks.

Findings

WPQSym forms a Hopf subalgebra of WCQSym.

WPQSym is a Hopf quotient algebra of WCQSym.

WPQSym embeds as a Rota-Baxter subalgebra of WCQSym.

Abstract

In this paper, we construct the weak version of peak quasisymmetric functions inside the Hopf algebra of weak composition quasisymmetric functions (WCQSym) defined by Guo, Thibon and Yu. Weak peak quasisymmetric functions (WPQSym) are studied in several aspects. First we find a natural basis of WPQSym lifting Stembridge's peak functions. Then we confirm that WPQSym is a Hopf subalgebra of WCQSym by giving explicit multiplication, comultiplication and antipode formulas. By extending Stembridge's descent-to-peak maps, we also show that WPQSym is a Hopf quotient algebra of WCQSym. On the other hand, we prove that WPQSym embeds as a Rota-Baxter subalgebra of WCQSym, thus of the free commutative Rota-Baxter algebra of weight 1 on one generator. Moreover, WPQSym can also be a Rota-Baxter quotient algebra of WCQSym.