Free quasi-symmetric functions of arbitrary level

TL;DR

This paper introduces generalized free quasi-symmetric functions labeled by colored permutations, unifying various algebraic structures like descent algebras, parking functions, and non-crossing partitions through Hopf algebra frameworks.

Contribution

It develops new analogues of Hopf algebras of free quasi-symmetric functions for arbitrary levels, linking them to colored combinatorial objects and wreath product structures.

Findings

Recovered descent algebras for wreath products in a simple way

Constructed Hopf algebras of colored parking functions and non-crossing partitions

Unified various combinatorial Hopf algebras under a common framework

Abstract

We introduce analogues of the Hopf algebra of Free quasi-symmetric functions with bases labelled by colored permutations. As applications, we recover in a simple way the descent algebras associated with wreath products $\Gamma\wr\SG_n$ and the corresponding generalizations of quasi-symmetric functions. Also, we obtain Hopf algebras of colored parking functions, colored non-crossing partitions and parking functions of type $B$.