Superspace coinvariants for wreath products

Let $\Omega$ be the superspace ring of regular differential forms on the affine space $\mathbb{C}^n$.

If $G \subseteq GL_n(\mathbb{C})$ is a complex reflection group, the {\em $G$-superspace coinvariant ring} is the quotient $SR_G := \Omega_n/SI_G$ where $SI_G \subseteq \Omega$ is the ideal generated by $G$-invariants with vanishing constant term.

We study this ring when $G = \mathbb{Z}_r \wr \mathfrak{S}_n$ is the group of $r$-colored permutation matrices.

We prove a conjecture of Sagan and Swanson on a monomial basis for $SR_G$ and give an Operator Theorem description of its inverse system.

We also give a combinatorial model for the ungraded and exterior-graded structure of $SR_G$ as a $G$-module.