A uniform generalization of some combinatorial Hopf algebras

TL;DR

This paper introduces a unified framework extending classical combinatorial Hopf algebras to all finite Coxeter systems, revealing new connections with 0-Hecke algebra representations and dual modules in types B and D.

Contribution

It generalizes key Hopf algebras to a broader class of Coxeter systems and explores their representation-theoretic relationships, especially in types B and D.

Findings

Unified Hopf algebra structures for all finite Coxeter systems

Connections established between these algebras and 0-Hecke algebra representations

Dual modules and comodules identified in types B and D

Abstract

We generalize the Hopf algebras of free quasisymmetric functions, quasisymmetric functions, noncommutative symmetric functions, and symmetric functions to certain representations of the category of all finite Coxeter systems and its dual category. We investigate their connections with the representation theory of 0-Hecke algebras of finite Coxeter systems. Restricted to type B and D we obtain dual graded modules and comodules over the corresponding Hopf algebras in type A.