Descent representations and colored quasisymmetric functions

TL;DR

This paper extends the concept of descent representations and colored quasisymmetric functions from symmetric groups to colored permutation groups, introducing new combinatorial objects and formulas.

Contribution

It introduces colored zigzag shapes and proves their connection to colored descent representations, generalizing previous results to multiple colors.

Findings

Colored zigzag shapes correspond to colored descent representations.

The paper generalizes MaMahon's alternating formula to the colored setting.

Colored quasisymmetric functions are shown to be symmetric and Schur-positive.

Abstract

The quasisymmetric generating function of the set of permutations whose inverses have a fixed descent set is known to be symmetric and Schur-positive. The corresponding representation of the symmetric group is called the descent representation. In this paper, we provide an extension of this result to colored permutation groups, where Gessel's fundamental quasisymmetric functions are replaced by Poirier's colored quasisymmetric functions. For this purpose, we introduce a colored analogue of zigzag shapes and prove that the representations associated with these shapes coincide with colored descent representations studied by Adin, Brenti and Roichman in the case of two colors and Bagno and Biagioli in the general case. Additionally, we provide a colored analogue of MaMahon's alternating formula which expresses ribbon Schur functions in the basis of complete homogeneous symmetric functions.