The simple multiplicity conjecture for fusion rules of the extended algebra

Let $\mathfrak W_{n,\ell}$ be the extended algebra, with atypical modules $\mathsf{A}_{n',\ell'}^{n,\ell}$ and typical modules $\mathsf{T}_{n”,e”}^{n,\ell}$. The fusion product is denoted by $\times_f$, and $a$ is the parameter appearing in the proposed atypical–typical fusion rule.

Simple multiplicity conjecture. The fusion rules of $\mathfrak W_{n,\ell}$ are

$$\begin{split} \mathsf{A}_{n',\ell'}^{n,\ell}\times_f\mathsf{A}_{n”,\ell”}^{n,\ell}&=\mathsf{A}_{n'+n”,\ell'+\ell”}^{n,\ell},\\ \mathsf{A}_{n',\ell'}^{n,\ell}\times_f\mathsf{T}_{n”,e”}^{n,\ell}&=\mathsf{T}_{n'+n”+a\ell',\ell'+e”}^{n,\ell}. \end{split}$$

The authors motivate this by noting that the underlying fusion rules determine which fusion coefficients are nonzero, while their precise multiplicities are not known; the conjecture asserts that these multiplicities are as simple as possible.