Jacobi–Trudi determinant formula for projective covers

Suppose that $\mathfrak{g}$ is of type $B_n$, $C_n$ or $D_{n+1}$ and that $\lambda\in P^+$ satisfies $\lambda(h_i)=0$ for $i\ge n$. Let $\Psi_\lambda$ be the root subset defined from $i_\lambda=\max\{i\in I:\lambda(h_i)>0\}$, and let $P(\lambda,0)^{\Gamma(\lambda,\Psi_\lambda)}$ denote the corresponding projective object. Using the Jacobi–Trudi determinant $\boldsymbol{h}_\lambda$ defined from the characters of the fundamental representations, Jacobi–Trudi projective-cover conjecture.

$$\operatorname{ch}P(\lambda,0)^{\Gamma(\lambda,\Psi_\lambda)}=\boldsymbol{h}_\lambda.$$

Equivalently, with the coefficients $c^\lambda_{\nu,s}$ and indexing set $\Gamma(\lambda,\Psi_\lambda)$ from the paper,

$$\sum_{(\nu,s)\in\Gamma(\lambda,\Psi_\lambda)}(-1)^s c^\lambda_{\nu,s}\boldsymbol{h}_\nu=\operatorname{ch}V(\lambda).$$

This conjecture combines the projective-cover problem with the Nakai–Nakanishi character conjecture; it is verified only in certain cases in the paper and remains open in the stated generality.