Multigraded sign-character conjecture for R_n^{(1,3)}

Prove the proposed multigraded formula for the sign-character multiplicity of the (1,3)-bosonic-fermionic coinvariant ring R_n^{(1,3)}, namely \(\langle \Frob(R^{(1,3)};q;u,v,w),s_{(1^n)}\rangle=\sum_{k,\ell,d\geq0}u^kv^\ell w^d q^{\binom{n-d-k-\ell}{2}}\qbinom{n-1-d}{\ell}_q\qbinom{n-1-k}{d}_q\qbinom{n-1-\ell}{k}_q\) for every n\geq1.

Background

The paper derives a conditional upper bound for the sign-character multiplicity in R_n^{(1,3)} using the specialized Theta conjecture and reports computational agreement for n\leq5.

Based on these data, the author proposes the displayed multigraded identity. The paper notes that, if the specialized Theta conjecture and Bergeron’s dimension conjecture both hold, the resulting multiplicity would attain the upper bound and prove this conjecture.