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).

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).

A fusion ring $\mathcal{F}$ (see here p 28 for more details) 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 $\sum d_i^2$.
$\mathcal{F}$ is simple if it has no non-trivial fusion subring.

The Grothendieck ring of a fusion category (see here for more details) is a fusion ring.
$\mathcal{F}$ is categorifiable if it is the Grothendieck ring of a fusion category.

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.

For all ranks $<9$ and all dimensions $<210$, all the integral simple fusion rings are trivial (i.e. isomorphic to a $\mathcal{F}(G)$), 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).

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).