Superspace coinvariants for wreath products

Abstract

Let $\hat{I}\text{c◯}$ be the superspace ring of regular differential forms on the affine space ${\mathbb{C}}^n$. If $G\subseteq G{L}_n(\mathbb{C})$ is a complex reflection group, the {\em $G$-superspace coinvariant ring} is the quotient $S{R}_G:=\hat{I}{\text{c◯}}_n/S{I}_G$ where $S{I}_G\subseteq \hat{I}\text{c◯}$ 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 $S{R}_G$ and give an Operator Theorem description of its inverse system. We also give a combinatorial model for the ungraded and exterior-graded structure of $S{R}_G$ as a $G$-module.

46 pages