Theta conjecture for the two-boson, two-fermion coinvariant ring

Establish that, for every integer n≥1, the multigraded Frobenius series of the two-boson, two-fermion coinvariant ring R_n^{(2,2)} satisfies \[\Frob(R_n^{(2,2)};q,t;u,v)=\sum_{k+\ell<n}u^kv^\ell\Theta_{e_k}\Theta_{e_\ell}\nabla e_{n-k-\ell}.\]

Background

The paper studies bosonic-fermionic coinvariant rings R_n{(k,j)}, whose Frobenius series encode the multigraded symmetric-group representations carried by the rings. The cited Theta conjecture proposes an explicit expression for the Frobenius series of R_n{(2,2)} in terms of Theta operators, the nabla operator, and elementary symmetric functions.

This conjecture is used as background for a specialization at t=0 concerning R_n{(1,2)}. The paper does not prove the general R_n{(2,2)} identity; instead, it invokes it as a conjectural result in deriving conditional information about hook characters and the sign character of R_n{(1,3)}.

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 of D'Adderio, Iraci, and Vanden Wyngaerd, recalled in Section 1 of the paper