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

Jean-Christophe Aval (1)

1 LaBRI - Laboratoire Bordelais de Recherche en Informatique

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 $\Q[x_1,\dots,x_n]$, 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 Gröbner 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.

Domaines: Combinatoire [math.CO]

Jean-Christophe Aval. Quasi-invariant and super-coinvariant polynomials for the generalized symmetric group. Formal Power Series and Algebraic Combinatorics, 2003, Linköping, Sweden. ⟨hal-00185530⟩