A proof of the fermionic Theta coinvariant conjecture

Let $(x_1, \dots, x_n, y_1, \dots, y_n)$ be a list of $2n$ commuting variables, $(θ_1, \dots, θ_n, ξ_1, \dots, ξ_n)$ be a list of $2n$ anticommuting variables, and $\mathbb{C}[X_n, Y_n] \otimes \wedge \{Θ_n, Ξ_n\}$ be the algebra generated by these variables. D'Adderio, Iraci, and Vanden Wyngaerd introduced the {\em Theta operators} on the ring of symmetric functions and used them to conjecture a formula for the quadruply-graded $\mathfrak{S}_n$-isomorphism type of $\mathbb{C}[X_n,Y_n] \otimes \wedge \{Θ_n, Ξ_n\}/I$ where $I$ is the ideal generated by $\mathfrak{S}_n$-invariants with vanishing constant term. We prove their conjecture in the `purely fermionic setting' obtained by setting the commuting variables equal $x_i, y_i$ equal to zero.

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