Free quasi-symmetric functions and descent algebras for wreath products, and noncommutative multi-symmetric functions

TL;DR

This paper develops new algebraic structures related to colored permutations, wreath products, and multi-symmetric functions, generalizing classical symmetric function theory and introducing internal products and descent algebras for these objects.

Contribution

It introduces analogs of free quasi-symmetric functions for colored permutations and constructs generalized descent algebras for wreath products, extending classical symmetric function frameworks.

Findings

Constructed Hopf algebras with bases labeled by colored permutations

Defined internal products for wreath product-related functions

Extended constructions to colored parking functions and non-crossing partitions

Abstract

We introduce analogs of the Hopf algebra of Free quasi-symmetric functions with bases labelled by colored permutations. When the color set is a semigroup, an internal product can be introduced. This leads to the construction of generalized descent algebras associated with wreath products $\Gamma\wr\SG_n$ and to the corresponding generalizations of quasi-symmetric functions. The associated Hopf algebras appear as natural analogs of McMahon's multisymmetric functions. As a consequence, we obtain an internal product on ordinary multi-symmetric functions. We extend these constructions to Hopf algebras of colored parking functions, colored non-crossing partitions and parking functions of type B.