Theta conjecture for the (2,2)-bosonic-fermionic coinvariant ring

Establish the multigraded Frobenius-series identity for the (2,2)-bosonic-fermionic coinvariant ring R_n^{(2,2)} given by \(\Frob(R_n^{(2,2)};q,t;u,v)=\sum_{k+\ell<n}u^kv^\ell\Theta_{e_k}\Theta_{e_\ell}\nabla e_{n-k-\ell}\) for every n\geq1.

Background

The paper recalls a conjecture of D’Adderio, Iraci, and Vanden Wyngaerd concerning the multigraded Frobenius series of the coinvariant ring with two sets of bosonic variables and two sets of fermionic variables. It expresses the Frobenius series through Theta operators and the nabla operator.

The conjecture is used as background for conditional results about hook characters and an upper bound for the sign-character multiplicity of the (1,3)-bosonic-fermionic coinvariant ring. It is not proved in the paper.