Generalized coinvariant algebras for wreath products

Abstract

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 $G_n$.