Quasi-invariant and super-coinvariant polynomials for the generalized symmetric group

TL;DR

This paper extends the theory of super-coinvariant polynomials to the generalized symmetric group, defining a new quasi-symmetrizing action and characterizing the structure and dimension of the associated polynomial spaces.

Contribution

It introduces a quasi-symmetrizing action of the generalized symmetric group and describes a Gröbner basis for the related ideal, revealing the dimension of super-coinvariant polynomials.

Findings

Dimension of super-coinvariant space is m^n times the nth Catalan number.

A Gröbner basis for the quasi-invariant ideal is explicitly described.

The work generalizes previous symmetric group results to wreath products.

Abstract

The aim of this work is to extend the study of super-coinvariant polynomials, to the case of the generalized symmetric group $G_{n,m}$, defined as the wreath product $C_m\wr\S_n$ of the symmetric group by the cyclic group. We define a quasi-symmetrizing action of $G_{n,m}$ on $\Q[x_1,...,x_n]$, analogous to those defined by Hivert in the case of $\S_n$. The polynomials invariant under this action are called quasi-invariant, and we define super-coinvariant polynomials as polynomials orthogonal, with respect to a given scalar product, to the quasi-invariant polynomials with no constant term. Our main result is the description of a Gr\"obner basis for the ideal generated by quasi-invariant polynomials, from which we dedece that the dimension of the space of super-coinvariant polynomials is equal to $m^n C_n$ where $C_n$ is the $n$-th Catalan number.