Multi-quasisymmetric functions with semigroup exponents, Hopf algebras and Rota-Baxter algebras

TL;DR

This paper generalizes quasisymmetric functions using semigroup exponents and multiple variables, establishing new Hopf algebra structures linked to Rota-Baxter algebras, expanding the algebraic framework for symmetric function theory.

Contribution

It introduces multi-quasisymmetric functions with semigroup exponents and constructs associated Hopf algebra structures, extending prior work on quasisymmetric functions and Rota-Baxter algebras.

Findings

Defined the space of multi-quasisymmetric functions $ ext{MQSym}$ and its basis functions.

Established a Hopf algebra structure on $ ext{MQSym}^E$ with semigroup exponents.

Proved the isomorphism between free Rota-Baxter algebras and scalar extensions of $ ext{MQSym}^E$.

Abstract

Many years ago, G.-C.~Rota discovered a close connection between symmetric functions and Rota-Baxter algebras, and proposed to study generalizations of symmetric functions in the framework of Rota-Baxter algebras. Guided by this proposal, quasisymmetric functions from weak composition (instead of just compositions) were obtained from free Rota-Baxter algebras on one generator. This paper aims to generalize this approach to free Rota-Baxter algebras on multiple generators in order to obtain further generalizations of quasisymmetric functions. For this purpose and also for its independent interest, the space $\mathrm{MQSym}$ of quasisymmetric functions on multiple sequences of variables is defined, generalizing quasisymmetric functions and diagonally quasisymmetric functions of Aval, Bergeron and Bergeron. Linear bases of such multi-quasisymmetric functions are given by monomial multi-quasisymmetric functions and fundamental multi-quasisymmetric functions, the latter recover the fundamental $G^m$-quasisymmetric functions of Aval and Chapoton. Next introduced is the even more general notion of multi-quasisymmetric functions $\mathrm{MQSym}^E$ with exponents in a semigroup $E$, which also generalizes the quasisymmetric functions with semigroup exponents in a recent work. Through this approach, a natural Hopf algebraic structure is obtained on $\mathrm{MQSym}^E$. Finally, in support of Rota's proposal, the free commutative unitary Rota-Baxter algebra on a finite set is shown to be isomorphic to a scalar extension of $\mathrm{MQSym}^E$, a fact which in turn equips the free Rota-Baxter algebra with a Hopf algebra structure.