Fermionic Diagonal Coinvariants

Let $W$ be a complex reflection group of rank $n$ acting on its reflection representation $V \cong \mb{C}^n$. The doubly graded action of $W$ on the exterior algebra $\wedge (V \oplus V^)$ induces an action on the quotient by the ideal generate by $W$-invariants with vanishing constant term $\FDR_W = \wedge (V \oplus V^) / \langle \wedge (V \oplus V^*)^W_{+} \rangle$. We describe the bi-graded $W$-module structure of $\FDR_W$ and introduce a variant of Motzkin paths that descends to the standard monomial basis of $\FDR_W$ with respect to certain term order. The top degree of $\FDR_W$ exhibits the Narayana refinement of Catalan numbers. When $W = S_n$, the symmetric group, $\FDR_{S_n} \cong R_{n,0,2}$, where $R_{n,0,2}$ is the special case of the Boson-Fermionic diagonal coinvariants with two sets of Fermionic variables. In this case, the $(i,j)$-th degree component is a difference of Kronecker product of two hook Schur functions. In addition we consider a module $M_{n,m}$ spanned by $m$-ary strings of length $n$. When $m = 2$, as a vector space, $M_{n,2} \cong \mb{C}[X_n] / \langle x_1^2, \ldots, x_n^2 \rangle$. The trivial component of $\dr_n \otimes M_{n,2}$ is a weighted sum of $q,t$-Narayana numbers which is a different $q,t$-Catalan number than the alternant of $\dr_n$. At $t = 1$, the trivial component equals the inversion generating function for $321$-avoiding permutations.