Is there an integral fusion category of global dimension $210$, such that the simple objects have dimensions $\{1,5,5,5,6,7,7\}$ and the following fusion matrices?
$\small{\begin{smallmatrix}
1 & 0 & 0 & 0& 0& 0& 0 \\
0 & 1 & 0 & 0& 0& 0& 0 \\
0 & 0 & 1 & 0& 0& 0& 0 \\
0 & 0 & 0 & 1& 0& 0& 0 \\
0 & 0 & 0 & 0& 1& 0& 0 \\
0 & 0 & 0 & 0& 0& 1& 0 \\
0 & 0 & 0 & 0& 0& 0& 1
\end{smallmatrix} ,
\begin{smallmatrix}
0 & 1 & 0 & 0& 0& 0& 0 \\
1 & 1 & 0 & 1& 0& 1& 1 \\
0 & 0 & 1 & 0& 1& 1& 1 \\
0 & 1 & 0 & 0& 1& 1& 1 \\
0 & 0 & 1 & 1& 1& 1& 1 \\
0 & 1 & 1 & 1& 1& 1& 1 \\
0 & 1 & 1 & 1& 1& 1& 1
\end{smallmatrix} ,
\begin{smallmatrix}
0 & 0 & 1 & 0& 0& 0& 0 \\
0 & 0 & 1 & 0& 1& 1& 1 \\
1 & 1 & 1 & 0& 0& 1& 1 \\
0 & 0 & 0 & 1& 1& 1& 1 \\
0 & 1 & 0 & 1& 1& 1& 1 \\
0 & 1 & 1 & 1& 1& 1& 1 \\
0 & 1 & 1 & 1& 1& 1& 1
\end{smallmatrix} ,
\begin{smallmatrix}
0 & 0 & 0 & 1& 0& 0& 0 \\
0 & 1 & 0 & 0& 1& 1& 1 \\
0 & 0 & 0 & 1& 1& 1& 1 \\
1 & 0 & 1 & 1& 0& 1& 1 \\
0 & 1 & 1 & 0& 1& 1& 1 \\
0 & 1 & 1 & 1& 1& 1& 1 \\
0 & 1 & 1 & 1& 1& 1& 1
\end{smallmatrix} ,
\begin{smallmatrix}
0 & 0 & 0 & 0& 1& 0& 0 \\
0 & 0 & 1 & 1& 1& 1& 1 \\
0 & 1 & 0 & 1& 1& 1& 1 \\
0 & 1 & 1 & 0& 1& 1& 1 \\
1 & 1 & 1 & 1& 1& 1& 1 \\
0 & 1 & 1 & 1& 1& 2& 1 \\
0 & 1 & 1 & 1& 1& 1& 2
\end{smallmatrix} ,
\begin{smallmatrix}
0 & 0 & 0 & 0& 0& 1& 0 \\
0 & 1 & 1 & 1& 1& 1& 1 \\
0 & 1 & 1 & 1& 1& 1& 1 \\
0 & 1 & 1 & 1& 1& 1& 1 \\
0 & 1 & 1 & 1& 1& 2& 1 \\
1 & 1 & 1 & 1& 2& \color{purple}{1}& \color{purple}{2} \\
0 & 1 & 1 & 1& 1& \color{purple}{2}& \color{purple}{2}
\end{smallmatrix} ,
\begin{smallmatrix}
0 & 0 & 0 & 0& 0& 0& 1 \\
0 & 1 & 1 & 1& 1& 1& 1 \\
0 & 1 & 1 & 1& 1& 1& 1 \\
0 & 1 & 1 & 1& 1& 1& 1 \\
0 & 1 & 1 & 1& 1& 1& 2 \\
0 & 1 & 1 & 1& 1& \color{purple}{2}& \color{purple}{2} \\
1 & 1 & 1 & 1& 2& \color{purple}{2}& \color{purple}{1}
\end{smallmatrix}}$
or also the same rules with a little $\color{purple}{\text{variation}}$ for the 7-dim. simple objects (and mult. 3 instead of 2):
$\small{ \begin{smallmatrix}
1 & 0 & 0 & 0& 0& 0& 0 \\
0 & 1 & 0 & 0& 0& 0& 0 \\
0 & 0 & 1 & 0& 0& 0& 0 \\
0 & 0 & 0 & 1& 0& 0& 0 \\
0 & 0 & 0 & 0& 1& 0& 0 \\
0 & 0 & 0 & 0& 0& 1& 0 \\
0 & 0 & 0 & 0& 0& 0& 1
\end{smallmatrix} ,
\begin{smallmatrix}
0 & 1 & 0 & 0& 0& 0& 0 \\
1 & 1 & 0 & 1& 0& 1& 1 \\
0 & 0 & 1 & 0& 1& 1& 1 \\
0 & 1 & 0 & 0& 1& 1& 1 \\
0 & 0 & 1 & 1& 1& 1& 1 \\
0 & 1 & 1 & 1& 1& 1& 1 \\
0 & 1 & 1 & 1& 1& 1& 1
\end{smallmatrix} ,
\begin{smallmatrix}
0 & 0 & 1 & 0& 0& 0& 0 \\
0 & 0 & 1 & 0& 1& 1& 1 \\
1 & 1 & 1 & 0& 0& 1& 1 \\
0 & 0 & 0 & 1& 1& 1& 1 \\
0 & 1 & 0 & 1& 1& 1& 1 \\
0 & 1 & 1 & 1& 1& 1& 1 \\
0 & 1 & 1 & 1& 1& 1& 1
\end{smallmatrix} ,
\begin{smallmatrix}
0 & 0 & 0 & 1& 0& 0& 0 \\
0 & 1 & 0 & 0& 1& 1& 1 \\
0 & 0 & 0 & 1& 1& 1& 1 \\
1 & 0 & 1 & 1& 0& 1& 1 \\
0 & 1 & 1 & 0& 1& 1& 1 \\
0 & 1 & 1 & 1& 1& 1& 1 \\
0 & 1 & 1 & 1& 1& 1& 1
\end{smallmatrix} ,
\begin{smallmatrix}
0 & 0 & 0 & 0& 1& 0& 0 \\
0 & 0 & 1 & 1& 1& 1& 1 \\
0 & 1 & 0 & 1& 1& 1& 1 \\
0 & 1 & 1 & 0& 1& 1& 1 \\
1 & 1 & 1 & 1& 1& 1& 1 \\
0 & 1 & 1 & 1& 1& 2& 1 \\
0 & 1 & 1 & 1& 1& 1& 2
\end{smallmatrix} ,
\begin{smallmatrix}
0 & 0 & 0 & 0& 0& 1& 0 \\
0 & 1 & 1 & 1& 1& 1& 1 \\
0 & 1 & 1 & 1& 1& 1& 1 \\
0 & 1 & 1 & 1& 1& 1& 1 \\
0 & 1 & 1 & 1& 1& 2& 1 \\
1 & 1 & 1 & 1& 2& {\color{purple}{0}}& {\color{purple}{3}} \\
0 & 1 & 1 & 1& 1& {\color{purple}{3}}& {\color{purple}{1}}
\end{smallmatrix} ,
\begin{smallmatrix}
0 & 0 & 0 & 0& 0& 0& 1 \\
0 & 1 & 1 & 1& 1& 1& 1 \\
0 & 1 & 1 & 1& 1& 1& 1 \\
0 & 1 & 1 & 1& 1& 1& 1 \\
0 & 1 & 1 & 1& 1& 1& 2 \\
0 & 1 & 1 & 1& 1& {\color{purple}{3}}& {\color{purple}{1}} \\
1 & 1 & 1 & 1& 2& {\color{purple}{1}}& {\color{purple}{2}}
\end{smallmatrix}}$
Remark: these are the fusion matrices of the first two simple integral non-trivial fusion rings.
Note that $210 = 2.3.5.7$ and that these matrices are self-dual and irreducibles. They also commute.
By arXiv:0809.3031 proposition 9.11, if such integral fusion categories exist, they couldn't be "weakly group theoretical", and by arXiv:1208.0840 corollary 6.16, they would be abelian but not braided.
Thank you to Eric Rowell and Leonid Vainermann for these references.
Also thanks to Dave Penneys for asking Eric.
The proof that such a fusion category $\mathcal{C}$ can't be braided is the following completed argument:
If it's braided, then it can be non-degenerated (i.e. $\mathcal{C}′=Vec$) or degenerated:
- If it's non-degenerated then the contradiction follows by the corollary 6.16 cited.
- Else if it's degenerated, then by simplicity $\mathcal{C}′=\mathcal{C}$, so $\mathcal{C}$ is symmetric, and by Deligne, $\mathcal{C}≃Rep(G)$ as fusion category (without considering the symmetric structures), with $G$ a finite simple group, contradiction (because there is no simple group of order $210$).
Edit about the original motivation (July 2013):
These matrices are naturally came from my will of classifying the cyclic subfactors:
The first case I consider is "depth 2, irreducible, finite index", i.e. finite dimensional C*-Hopf algebras (also called Kac algebras). The first question to answer is:
Are there non-trivial cyclic Kac algebras ? If so, the first example is certainly maximal.
Now a Kac algebra gives a unitary integral fusion category, so I have written an algorithm investigating all the integral fusion rings of a restrictive class containing necessarily those related to the non-trivial maximal Kac algebras. There are finitely many possibilities for each dimension.
Edit (June 2014):
I had also discovered eight fusion rings of global dimensions 360 and 660, with simple objects of dimensions $\{1,5,5,8,8,9,10\}$ and $\{1,5,5,10,10,11,12,12\}$. Two of them come from the simple groups $A_6$ and $A_1(11)$, the six others are new (see the fusion rules here).
This post imported from StackExchange MathOverflow at 2014-09-30 08:34 (UTC), posted by SE-user Sébastien Palcoux
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