A combinatorial model for the fermionic diagonal coinvariant ring

Let $\Theta_n = (\theta_1, \dots, \theta_n)$ and $\Xi_n = (\xi_1, \dots, \xi_n)$ be two lists of $n$ variables and consider the diagonal action of $\mathfrak{S}_n$ on the exterior algebra $\wedge \{ \Theta_n, \Xi_n \}$ generated by these variables. Jongwon Kim and Rhoades defined and studied the fermionic diagonal coinvariant ring $FDR_n$ obtained from $\wedge \{ \Theta_n, \Xi_n \}$ by modding out by the $\mathfrak{S}_n$-invariants with vanishing constant term. In joint work with Rhoades we gave a basis for the maximal degree components of this ring where the action of $\mathfrak{S}_n$ could be interpreted combinatorially via noncrossing set partitions. This paper will do similarly for the entire ring, although the combinatorial interpretation will be limited to the action of $\mathfrak{S}_{n-1} \subset \mathfrak{S}_n$. The basis will be indexed by a certain class of noncrossing partitions.