广义对称群的拟不变和超共不变多项式
Quasi-invariant and super-coinvariant polynomials for the generalized symmetric group
摘要：
本文的目的是将超共不变多项式的研究扩展到广义对称群$G_{n,m}$的情形，定义为对称群的圈积$C_m\wr§_n$与循环群的环积.我们定义了$G_{n,m}$对$\Q[x_1,...,x_n]$的拟对称化作用，类似于Hivert在$§_n$的情况下定义的作用。在此作用下不变的多项式称为准不变多项式，我们定义超余不变多项式为正交多项式，关于一个给定的标量积，准不变多项式没有常数项。我们的主要结果是描述了拟不变多项式生成的理想的Gröbner基，由此我们确定了超共不变多项式空间的维数等于$m^n C_n$，其中$C_n$是第$n$ -阶Catalan数.
英文摘要：
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 $C_m\wr§_n$ 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 $§_n$. 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.
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