4
$\begingroup$

A fusion ring $\mathcal{F}$ (see here p 28) is integral if the Perron-Frobenius dimension $d(h_i)$ of its basic elements $\{h_1,...,h_r\}$, are integers. Its rank is $r$ and its dimension is $\sum d(h_i)^2$.
$\mathcal{F}$ is simple if it has no non-trivial fusion subring.

$\mathcal{F}$ is categorifiable if it is the Grothendieck ring of a fusion category (see here).
(The pentagonal equation has a solution)

Let $Rep(G)$ be the fusion category of representations of a finite group $G$.
The Grothendieck ring $\mathcal{F}(G)$ of $Rep(G)$, is simple iff $G$ is a simple group.

Every integral simple fusion ring of rank $<9$ and dimension $<210$, are trivial (i.e. $\simeq \mathcal{F}(G)$), and so are categorifiable.

Question : Is an integral simple fusion ring, always categorifiable ?

The first non-trivial simple integral fusion ring I know, is rank $7$ and dimension $210$ (see here).

Improve this question
$\endgroup$
2
  • 1
    $\begingroup$ I am not an expert on this topic but in Dave Penneys' answer to this other thread it is mentioned that it is not known whether rank 4 fusion rings are categorifiable. $\endgroup$
    – Juan Bermejo Vega
    Commented Aug 18, 2015 at 9:45
  • $\begingroup$ Here is a simple integral fusion ring ruled out from unitary categorification. This atlas provides many simple integral fusion rings without any categorification. So the answer is no. $\endgroup$
    – Sebastien Palcoux
    Commented Oct 24, 2020 at 16:28

0

Reset to default

You must log in to answer this question.