[Paper Review] Free quasi-symmetric functions and descent algebras for wreath products, and noncommutative multi-symmetric functions

Jean-Christophe Novelli, Jean‐Yves Thibon|arXiv (Cornell University)|Jun 23, 2008

Advanced Combinatorial Mathematics23 references4 citations

TL;DR

This paper introduces level-$\ell$ analogs of free quasi-symmetric functions and noncommutative symmetric functions, using colored permutations and wreath products $\Gamma \wr \mathfrak{S}_n$. It constructs generalized descent algebras for wreath products, establishes internal products, and extends these to colored parking functions and non-crossing partitions. The key contribution is a noncommutative analog of McMahon's multi-symmetric functions, with a dual structure to Poirier's quasi-symmetric functions and an internal product on ordinary multi-symmetric functions via commutative image.

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 $Γ\wr\SG_n$ 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.

Motivation & Objective

• To develop a noncommutative analog of McMahon's multi-symmetric functions using colored permutations and wreath products $\Gamma \wr \mathfrak{S}_n$.
• To construct generalized descent algebras for wreath products via level-$\ell$ free quasi-symmetric functions and establish their internal product structure.
• To extend the theory to colored parking functions, non-crossing partitions of type $B$, and planar binary trees, revealing new algebraic and combinatorial structures.
• To derive a noncommutative internal product on ordinary multi-symmetric functions through the commutative image of the level-$\ell$ constructions.

Proposed method

• Introduces the Hopf algebra ${\bf FQSym}^{(\ell)}$ with bases indexed by $\ell$-colored permutations, generalizing the standard ${\bf FQSym}$.
• Constructs ${\bf Sym}^{(\ell)}$ as a Hopf subalgebra of ${\bf FQSym}^{(\ell)}$, dual to Poirier's quasi-symmetric functions.
• Defines an internal product on ${\bf Sym}^{(\ell)}$ using colored shifted operations and dendriform structures, generalizing Solomon's descent algebra.
• Applies a Cauchy formula to compute the dual of the Mantaci-Reutenauer algebra ${\rm MR}^{(\ell)}$ and derive its internal product.
• Introduces the Hopf algebra ${\bf PQSym}^{(\ell)}$ of colored parking functions and establishes its bidendriform/tridendriform structure.
• Uses noncommutative Lagrange inversion and colored tree enumeration to interpret Raney's functional equation combinatorially.

Experimental results

Research questions

• RQ1 How can the Hopf algebra of free quasi-symmetric functions be generalized to colored permutations to model wreath product symmetries?
• RQ2 What internal product structure arises in the level-$\ell$ generalization of noncommutative symmetric functions, and how does it relate to descent algebras for $\Gamma \wr \mathfrak{S}_n$?
• RQ3 Can the construction of multi-symmetric functions be lifted to a noncommutative setting with an internal product?
• RQ4 How do colored parking functions and non-crossing partitions of type $B$ fit into the framework of level-$\ell$ Hopf algebras?
• RQ5 What is the combinatorial interpretation of Raney's functional equation in the context of colored trees and noncommutative Lagrange inversion?

Key findings

• The Hopf algebra ${\bf FQSym}^{(\ell)}$ is constructed with bases indexed by $\ell$-colored permutations, admitting an internal product when the color set is a semigroup.
• The subalgebra ${\bf Sym}^{(\ell)}$ is isomorphic to the free product of $\ell$ copies of ${\bf Sym}$, and is dual to Poirier's quasi-symmetric functions.
• The homogeneous components of ${\bf Sym}^{(\ell)}$ carry an internal product, generalizing Solomon's descent algebra for wreath products $\Gamma \wr \mathfrak{S}_n$.
• The commutative image of ${\bf Sym}^{(\ell)}$ yields an internal product on ordinary multi-symmetric functions, extending McMahon's construction.
• The Mantaci-Reutenauer algebra ${\rm MR}^{(\ell)}$ arises as a Hopf subalgebra of ${\bf Sym}^{(\ell)}$, with its dual computed via a Cauchy formula.
• The Hopf algebra ${\bf PQSym}^{(\ell)}$ of colored parking functions admits a bidendriform/tridendriform structure and supports a noncommutative Lagrange inversion formula.