Recently (what I believe are) all multiplicity-free fusion categories up to rank 7 have been posted on the AnyonWiki. Most of the fusion rings belonging to these categories belong to one of the following classes:
- Finite group rings, e.g., cyclic groups of order $1,\ldots,7$, $D_3$, etc
- Representation rings of finite groups, e.g., $\text{Rep}(D_3),\text{Rep}(S_4)$, etc
- Fusion rings related to quantum groups, e.g., $\text{SU}(2)_{k}$ for $k=1,\ldots,6$, $\text{SO}(2N+1)_{2}$ for $N=1,\ldots,3$, etc
- Fusion rings related to subfactors, e.g., a ring which we called Pseudo $\text{PSU}(2)_6$ (for now)
- Extensions of the above, such as $\text{HI}(\mathbb{Z}_3)$, $\text{TY}(G)$, for all finite abelian groups $G$ with $|G|<7$
- Products of the above, e.g. $\mathbb{Z}_2 \times R$ for any fusion ring $R$ with rank $\leq 3$, etc
- Zestings of the above
- Adjoint fusion rings of rings related to quantum groups, e.g., $\text{Adj}(\text{SO}(11)_2)$, $\text{Adj}(\text{SO}(16)_2)$
There are a few fusion rings that I don't immediately recognize as any of the above, though. (Which doesn't mean they aren't of the form above)
So, I wondered if anyone recognizes some of the fusion rings below as being related to some known constructions. They are all categorifiable into a unitary fusion category, but none of the categories admit a braided structure.
- The Moore-Read fusion ring
This one is known from physics. It has Frobenius-Perron dimensions $(1,1,1,1,\sqrt{2},\sqrt{2})$ and the following multiplication table
$\begin{array}{|llllll|} \hline \mathbf{1} & \mathbf{2} & \mathbf{3} & \mathbf{4} & \mathbf{5} & \mathbf{6} \\ \mathbf{2} & \mathbf{1} & \mathbf{4} & \mathbf{3} & \mathbf{5} & \mathbf{6} \\ \mathbf{3} & \mathbf{4} & \mathbf{2} & \mathbf{1} & \mathbf{6} & \mathbf{5} \\ \mathbf{4} & \mathbf{3} & \mathbf{1} & \mathbf{2} & \mathbf{6} & \mathbf{5} \\ \mathbf{5} & \mathbf{5} & \mathbf{6} & \mathbf{6} & \mathbf{3}+\mathbf{4} & \mathbf{1}+\mathbf{2} \\ \mathbf{6} & \mathbf{6} & \mathbf{5} & \mathbf{5} & \mathbf{1}+\mathbf{2} & \mathbf{3}+\mathbf{4} \\ \hline \end{array}$
Is there any way that this ring fits in the groups above?
- A ring we called Pseudo $\text{Rep}(S_4)$
Its multiplication table looks a lot like that from $\text{Rep}(S_4)$. It is listed in this paper at page 11 but the paper also says: Model Unknown. Its Frobenius-Perron dimensions are $(1,1,2,3,3)$ and its multiplication table is the following.
$\begin{array}{|lllll|} \hline \mathbf{1} & \mathbf{2} & \mathbf{3} & \mathbf{4} & \mathbf{5} \\ \mathbf{2} & \mathbf{1} & \mathbf{3} & \mathbf{5} & \mathbf{4} \\ \mathbf{3} & \mathbf{3} & \mathbf{1}+\mathbf{2}+\mathbf{3} & \mathbf{4}+\mathbf{5} & \mathbf{4}+\mathbf{5} \\ \mathbf{4} & \mathbf{5} & \mathbf{4}+\mathbf{5} & \mathbf{2}+\mathbf{3}+\mathbf{4}+\mathbf{5} & \mathbf{1}+\mathbf{3}+\mathbf{4}+\mathbf{5} \\ \mathbf{5} & \mathbf{4} & \mathbf{4}+\mathbf{5} & \mathbf{1}+\mathbf{3}+\mathbf{4}+\mathbf{5} & \mathbf{2}+\mathbf{3}+\mathbf{4}+\mathbf{5} \\ \hline \end{array}$
- A ring called $\text{FR}^{7,1,2}_{3}$ of which I have no idea where it comes from. Its Frobenius-Perron dimensions are $(1,1,1,1,2,2,2)$ and it has the following multiplication table
$\begin{array}{|lllllll|} \hline \mathbf{1} & \mathbf{2} & \mathbf{3} & \mathbf{4} & \mathbf{5} & \mathbf{6} & \mathbf{7} \\ \mathbf{2} & \mathbf{1} & \mathbf{4} & \mathbf{3} & \mathbf{5} & \mathbf{6} & \mathbf{7} \\ \mathbf{3} & \mathbf{4} & \mathbf{2} & \mathbf{1} & \mathbf{7} & \mathbf{6} & \mathbf{5} \\ \mathbf{4} & \mathbf{3} & \mathbf{1} & \mathbf{2} & \mathbf{7} & \mathbf{6} & \mathbf{5} \\ \mathbf{5} & \mathbf{5} & \mathbf{7} & \mathbf{7} & \mathbf{1}+\mathbf{2}+\mathbf{6} & \mathbf{5}+\mathbf{7} & \mathbf{3}+\mathbf{4}+\mathbf{6} \\ \mathbf{6} & \mathbf{6} & \mathbf{6} & \mathbf{6} & \mathbf{5}+\mathbf{7} & \mathbf{1}+\mathbf{2}+\mathbf{3}+\mathbf{4} & \mathbf{5}+\mathbf{7} \\ \mathbf{7} & \mathbf{7} & \mathbf{5} & \mathbf{5} & \mathbf{3}+\mathbf{4}+\mathbf{6} & \mathbf{5}+\mathbf{7} & \mathbf{1}+\mathbf{2}+\mathbf{6} \\ \hline \end{array}$
it is related via the testing construction to this ring with the same dimensions but a reshuffle of some of its structure constants.
$\begin{array}{|lllllll|} \hline \mathbf{1} & \mathbf{2} & \mathbf{3} & \mathbf{4} & \mathbf{5} & \mathbf{6} & \mathbf{7} \\ \mathbf{2} & \mathbf{1} & \mathbf{4} & \mathbf{3} & \mathbf{5} & \mathbf{6} & \mathbf{7} \\ \mathbf{3} & \mathbf{4} & \mathbf{2} & \mathbf{1} & \mathbf{5} & \mathbf{7} & \mathbf{6} \\ \mathbf{4} & \mathbf{3} & \mathbf{1} & \mathbf{2} & \mathbf{5} & \mathbf{7} & \mathbf{6} \\ \mathbf{5} & \mathbf{5} & \mathbf{5} & \mathbf{5} & \mathbf{1}+\mathbf{2}+\mathbf{3}+\mathbf{4} & \mathbf{6}+\mathbf{7} & \mathbf{6}+\mathbf{7} \\ \mathbf{6} & \mathbf{6} & \mathbf{7} & \mathbf{7} & \mathbf{6}+\mathbf{7} & \mathbf{3}+\mathbf{4}+\mathbf{5} & \mathbf{1}+\mathbf{2}+\mathbf{5} \\ \mathbf{7} & \mathbf{7} & \mathbf{6} & \mathbf{6} & \mathbf{6}+\mathbf{7} & \mathbf{1}+\mathbf{2}+\mathbf{5} & \mathbf{3}+\mathbf{4}+\mathbf{5} \\ \hline \end{array}$