Theta conjecture for the diagonal two-boson–two-fermion coinvariant ring

Prove that the multigraded Frobenius series of the diagonal two-boson–two-fermion coinvariant ring R_n^{(2,2)} is given by the stated sum of Theta-operator and nabla-operator expressions for every n≥1.

Background

The paper recalls a conjectural formula for the multigraded Frobenius series of R_n{(2,2)}, the coinvariant ring with two sets of commuting variables and two sets of anticommuting variables. The proposed expression uses the Theta operators Θ{e_k}, Θ{e_ℓ}, the nabla operator, and the elementary symmetric functions e_{n-k-ℓ}.

This conjecture is used as background for a t=0 specialization concerning R_n{(1,2)}. The paper does not prove either conjectural identity; instead, a later conditional theorem derives hook-character formulas assuming the specialization.

References

We first recall the ``Theta conjecture'' of D'Adderio, Iraci, and Vanden Wyngaerd, which expresses the multigraded Frobenius series of $R_n{(2,2)}$ in terms of certain Theta operators and the nabla operator. For all $n \geq 1$, \Frob(R_n{(2,2)}; q,t;u,v) = \sum_{ k + \ell < n} uk v\ell\Theta_{e_k}\Theta_{e_\ell}\nabla e_{n-k-\ell}.

— The sign character of the triagonal fermionic coinvariant ring  (2501.09920 - Lentfer, 17 Jan 2025) in Conjecture 8.2, Section 1, cited from D'Adderio, Iraci, and Vanden Wyngaerd (2021)