Let $r$ be a positive integer and let $G_n$ be the reflection group of $n \times n$ monomial matrices whose entries are $r^{th}$ complex roots of unity and let $k \leq n$. We define and study two new graded quotients $R_{n,k}$ and $S_{n,k}$ of the polynomial ring $\mathbb{C}[x_1, \dots, x_n]$ in $n$ variables. When $k = n$, both of these quotients coincide with the classical coinvariant algebra attached to $G_n$. The algebraic properties of our quotients are governed by the combinatorial properties of $k$-dimensional faces in the Coxeter complex attached to $G_n$ (in the case of $R_{n,k}$) and $r$-colored ordered set partitions of $\{1, 2, \dots, n\}$ with $k$ blocks (in the case of $S_{n,k}$). Our work generalizes a construction of Haglund, Rhoades, and Shimozono from the symmetric group $\mathfrak{S}_n$ to the more general wreath products $G_n$.

中文翻译：

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花圈乘积的广义共变代数

令 $r$ 为正整数，令 $G_n$ 为 $n \times n$ 个单项矩阵的反射群，其条目为 $r^{th}$ 复数单位根，令 $k \leq n$。我们定义并研究了 $n$ 变量中多项式环 $\mathbb{C}[x_1, \dots, x_n]$ 的两个新的分级商 $R_{n,k}$ 和 $S_{n,k}$。当 $k = n$ 时，这两个商都与附加到 $G_n$ 的经典共变代数重合。我们商的代数性质由附加到 $G_n$（在 $R_{n,k}$ 的情况下）和 $r$-colored 有序集的 Coxeter 复数中 $k$-维面的组合性质控制$\{1, 2, \dots, n\}$ 与 $k$ 块​​的分区（在 $S_{n,k}$ 的情况下）。我们的工作将 Haglund、Rhoades 和 Shimozono 的构造从对称群 $\mathfrak{S}_n$ 推广到更一般的花环积 $G_n$。