Symmetric functions, noncommutative symmetric functions, and quasisymmetric functions

TL;DR

This paper surveys the rich algebraic structures of symmetric functions, exploring their noncommutative and quasisymmetric generalizations, and discusses open questions in the field.

Contribution

It provides an overview of the structures and properties of noncommutative symmetric functions and quasisymmetric functions, highlighting analogies and open problems.

Findings

Analysis of Hopf algebra structures

Identification of properties with good analogues

Discussion of open research questions

Abstract

This paper is concerned with two generalizations of the Hopf algebra of symmetric functions that have more or less recently appeared. The Hopf algebra of noncommutative symmetric functions and its dual, the Hopf algebra of quasisymmetric functions. The focus is on the incredibly rich structure of the Hopf algebra of symmetric functions and the question of which structures and properties have good analogues for the noncommutative symmetric functions and/or the quasisymmetric functions. This paper attempt to survey the ongoing investigations in this topic as dictated by the knowledge and interests of its author. There are many open questions that are discussed.