Fusion Rings Related to Affine Weyl Groups

The construction of the fusion ring of a quasi-rational CFT based on $\widehat{sl}(3{)}_k$ at generic level $k\in ̸\mathbb{Q}$ is reviewed. It is a commutative ring generated by formal characters, elements in the group ring $\mathbb{Z}[\tilde{W}]$ of the extended affine Weyl group $\tilde{W}$ of $\widehat{sl}(3{)}_k$. Some partial results towards the $\widehat{sl}(4{)}_k$ generalisation of this character ring are presented.