Racah–Speiser recursion for the non-standard modules

Let $V^{\Lambda_m}$ be the non-standard module at level $m$, let $P(V^{\Lambda_m}_-)$ and $P(V^{\Lambda_m}_+)$ denote its lowest- and highest-weight portions, and let $W_{\mathcal F ib}$ be the Weyl group of ${\mathcal F}ib$. For a weight $\mu$, write $Mult_\lambda(\mu)$ for its multiplicity, let $\det(w)$ be the Weyl-group determinant, and set

$$\rho=\lambda_1+\lambda_2.$$

Racah–Speiser recursion conjecture. For $m=\pm1,\pm2$, the weights of $P(V^{\Lambda_m}_-)$ follow

$$Mult_\lambda(\mu)=-\sum_{1\neq w\in W_{\mathcal F}ib}\det(w)\,Mult_\lambda\bigl(\mu+(w\rho-\rho)\bigr),$$

and the weights of $P(V^{\Lambda_m}_+)$ follow

$$Mult_\lambda(\mu)=-\sum_{1\neq w\in W_{\mathcal F}ib}\det(w)\,Mult_\lambda\bigl(\mu-(w\rho-\rho)\bigr).$$

The computed weight data for the non-standard modules does not follow the Kac–Peterson recursion and instead exhibits the stated Racah–Speiser pattern. The claim is presented as an observed conjectural recursion for levels $m=\pm1,\pm2$.