# Non-"weakly group theoretical" integral fusion categories?

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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
retagged Sep 30, 2014
I am interested to know how you got those matrices: where do they come from?

This post imported from StackExchange MathOverflow at 2014-09-30 08:34 (UTC), posted by SE-user André Henriques
Kac algebras are not completely given by a unitary fusion category. You can have two non isomorphic Kac algebras with the same tensor category of representation, for example, a Drinfeld twist deformation of a group algebra.

This post imported from StackExchange MathOverflow at 2014-09-30 08:34 (UTC), posted by SE-user César Galindo
@CésarGalindo : Do you confirm they have the same fusion category (not only the same fusion ring). Maybe the fusion categories of the Kac algebra and its dual, give completely the Kac algebras. It's true for the few twist deformation of group algebras I know, but I don't know if it's true in general, and you ?

This post imported from StackExchange MathOverflow at 2014-09-30 08:34 (UTC), posted by SE-user Sébastien Palcoux
Yes, twist deformations of finite dimensional Hopf algebras have the same category of representation, look for example arxiv.org/pdf/math/0107167.pdf A finite dimensional Hopf algebra is completely determined by their category of representation and its fiber functor.

This post imported from StackExchange MathOverflow at 2014-09-30 08:34 (UTC), posted by SE-user César Galindo
In the same flavor that Cérar's comment, see the paper of Etingof-Gelaki : Isocategorical Groups

This post imported from StackExchange MathOverflow at 2014-09-30 08:34 (UTC), posted by SE-user Sébastien Palcoux
New perspectives are considered in the post : Abelian subfactors, a relevant concept?

This post imported from StackExchange MathOverflow at 2014-09-30 08:34 (UTC), posted by SE-user Sébastien Palcoux

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