The proposed superspace model for colored faces of wreath products

Assume $r\geq 2$. Let $\Omega$ be the polynomial-exterior superspace and let $SI'_{n,r}\subseteq\Omega$ be the superspace ideal generated by

$$x_1^{ir}+\cdots+x_n^{ir}\quad (i=1,2,\ldots,n)$$

and

$$x_1^{(i-1)r+1}\theta_1+\cdots+x_n^{(i-1)r+1}\theta_n\quad (i=1,2,\ldots,n).$$

Let ${\mathfrak F}_{n,r}$ be the family of ${\mathbb{Z}}_r\wr{\mathfrak S}_n$-faces, namely colored ordered set partitions whose zero-block elements all have color $0$.

Proposed colored-face model. The ideal $SI'_{n,r}$ is stable under ${\mathbb{Z}}_r\wr{\mathfrak S}_n$, and there is a module isomorphism

$$\Omega/SI'_{n,r}\cong_{{\mathbb{Z}}_r\wr{\mathfrak S}_n}\mathbb{C}[{\mathfrak F}_{n,r}]\otimes\det.$$

The construction is presented as conjectural in the conclusion and is intended as a superspace analogue of the colored-face models for wreath products. Its validity and the required structural properties remain open in the source.