# Generalized coinvariant algebras for wreath products

## TL;DR

This paper introduces new graded quotients of polynomial rings associated with wreath product reflection groups, generalizing classical coinvariant algebras and linking algebraic structures to combinatorial objects like Coxeter complex faces and colored set partitions.

## Contribution

It defines and studies two new quotients, $R_{n,k}$ and $S_{n,k}$, extending the coinvariant algebra framework from symmetric groups to wreath products $G_n$, connecting algebraic and combinatorial properties.

## Key findings

- The quotients coincide with classical coinvariant algebras when $k=n$.
- Algebraic properties are governed by combinatorial structures such as Coxeter complex faces and colored set partitions.
- The work generalizes previous constructions from symmetric groups to 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 $G_n$.