Polyn\^omes quasi-invariants et super-coinvariants pour le groupe sym\'etrique g\'en\'eralis\'e

TL;DR

This paper generalizes classical results on symmetric and quasi-symmetric polynomials to the generalized symmetric group, describing invariants and ideal codimensions involving Catalan numbers and wreath products.

Contribution

It introduces a quasi-symmetrizing action of the generalized symmetric group and determines the invariants and ideal codimension for this broader setting.

Findings

Invariants form an ideal of codimension m^n times Catalan number C_n.

Generalizes classical symmetric polynomial results to wreath products.

Provides explicit description of invariants under the generalized action.

Abstract

A classical result of Artin states that the ideal generated by symmetric polynomials in $n$ variables is of codimension $n!$. The author, F. Bergeron and N. Bergeron have recently obtained a surprising analogous in the case of quasi-symmetric polynomials. In this case, the ideal is of codimension given by $C_n$, the $n$-th Catalan number. Quasi-symmetric polynomials are the invariants of a certain action of the symmetric group $S_n$ defined by F. Hivert. The aim of this work is to generalize these results to the wreath product $S_n\wr \Z_m$, also known as the generalized symmetric group $G\nm$. We first define a quasi-symmetrizing action of $G\nm$ on $\C[x_1,...,x_n]$, then obtain a description of the invariants and the codimension of the associated ideal, which is $m^n C_n$.