Conjectural radical presentations of fusion ideals for types B and D

Let $I_{\ell}(\mathfrak{g})$ be the fusion ideal in the representation ring of the simple Lie algebra $\mathfrak{g}$, let $\sqrt{I}$ denote the radical of an ideal $I$, and let $\check{h}(\mathfrak{g})$ be the dual Coxeter number. For type $B_r$ with $r\geq 3$, use the fundamental weight $\omega_1$; for type $D_r$ with $r\geq 4$, use the fundamental weight $\omega_d$ of minimum Dynkin index.

Conjectural radical presentation. For $\mathfrak{g}$ of type $B_r$ with $r\geq 3$,

$$\sqrt{\langle \chi_{(\ell+1)\omega_1},\chi_{(\ell+2)\omega_1},\ldots,\chi_{(\ell+\check{h}(\mathfrak{g})-1)\omega_1}\rangle}=\langle \chi_{(\ell+1)\omega_1},\chi_{(\ell+2)\omega_1},\ldots,\chi_{(\ell+\check{h}(\mathfrak{g})-1)\omega_1},\chi_{\ell\omega_1+\omega_r}\rangle.$$

For $\mathfrak{g}$ of type $D_r$ with $r\geq 4$,

$$\sqrt{\langle \chi_{(\ell+1)\omega_d},\chi_{(\ell+2)\omega_d},\ldots,\chi_{(\ell+\check{h}(\mathfrak{g})-1)\omega_d}\rangle}=\langle \chi_{(\ell+1)\omega_d},\chi_{(\ell+2)\omega_d},\ldots,\chi_{(\ell+\check{h}(\mathfrak{g})-1)\omega_d},\chi_{\ell\omega_1+\omega_{r-1}},\chi_{\ell\omega_1+\omega_r}\rangle.$$

These conjectures refine the theorem's radical inclusions by proposing explicit generators for the corresponding fusion ideals. The preceding text notes that the inclusions in parts (a) and (b) are equalities for types $B_r$, $D_r$, and $G_2$; the additional equalities for types $B_r$ and $D_r$ remain conjectural.