Generalized coinvariant algebras for $G(r,1,n)$ in the Stanley-Reisner setting

TL;DR

This paper introduces generalized coinvariant algebras for complex reflection groups, extending classical constructions to new quotient rings that relate to Stanley-Reisner theory and combinatorial algebra.

Contribution

It defines new quotient algebras isomorphic to previously studied coinvariant quotients for complex reflection groups, generalizing known algebraic structures.

Findings

Constructed quotients $R_{n,k}$ and $S_{n,k}$ for complex reflection groups.

Established isomorphisms between these quotients and Stanley-Reisner type algebras.

Extended classical coinvariant algebra concepts to broader algebraic and combinatorial settings.

Abstract

Let $r$ and $n$ be positive integers, let $G_n$ be the complex reflection group of $n \times n$ monomial matrices whose entries are $r^{\textrm{th}}$ roots of unity and let $0 \leq k \leq n$ be an integer. Recently, Haglund, Rhoades and Shimozono ($r=1$) and Chan and Rhoades ($r>1$) introduced quotients $R_{n,k}$ (for $r>1$) and $S_{n,k}$ (for $r \geq 1$) of the polynomial ring $\mathbb{C}[x_1,\ldots,x_n]$ in $n$ variables, which for $k=n$ reduce to the classical coinvariant algebra attached to $G_n$. When $n=k$ and $r=1$, Garsia and Stanton exhibited a quotient of $\mathbb{C}[\mathbf{y}_S]$ isomorphic to the coinvariant algebra, where $\mathbb{C}[\mathbf{y}_S]$ is the polynomial ring in $2^n-1$ variables whose variables are indexed by nonempty subsets $S \subseteq [n]$. In this paper, we will define analogous quotients that are isomorphic to $R_{n,k}$ and $S_{n,k}$.