A *fusion ring* is a finite dimensional complex space
$\mathbb{C}\mathcal{B}$ together with a distinguished basis
$\mathcal{B} = \{ h_1,...,h_r\}$ and fusion rules $ h_i \cdot h_j =
 \sum_k n_{ij}^kh_k $, with $n_{ij}^k \in \mathbb{N}_{\geq 0}$
satisfying:  
  - *Neutral*: $n_{1i}^j = n_{i1}^j = \delta_{ij}$        
  - *Dual*: $\forall i \  \exists!j $ (noted $i^*$) such that $n_{ij}^1>0$  
  - *Associativity*: $\sum_s n_{ij}^sn_{sk}^t = \sum_s n_{jk}^sn_{is}^t$   
  - *Frobenius-Perron reciprocity*: $n_{ij}^k = n_{i^*k}^j = n_{kj^*}^i$    
    
 *Remark*: $\mathbb{C}\mathcal{B}$ admits a structure of finite dimensional ${\rm C}^*$-algebra (take $h_i^* = h_{i^*}$).  
 *Frobenius-Perron theorem*: $\exists!$ $*$-homomorphism $d:\mathbb{C}\mathcal{B} \to \mathbb{C}$ with $d(\mathcal{B}) \subset
 (0,\infty)$.
 
  
The *rank* $r$ of the fusion ring $\mathbb{C}\mathcal{B}$ is the cardinal of $\mathcal{B}$.   
It is is called *integral* if every
 $d(h_i)$ is an integer.   
Its *Frobenius-Perron dimension* (FPdim) is $\sum d(h_i)^2$.     
It is of *Frobenius type* if every $d(h_i)$ divides FPdim$(\mathbb{C}\mathcal{B})$.   
It is *simple* if $r>1$ and for any fusion subring $\mathbb{C}\mathcal{S} \subseteq \mathbb{C}\mathcal{B}$ with $\mathcal{S} \subseteq \mathcal{B}$, then $\mathcal{S} = \{ h_1 \}$ or $\mathcal{B}$.

*Open problem*: Every fusion ring is of Frobenius type.

 *Remark*: The Grothendieck ring of a finite group $G$ is the ring generated by the irreducible complex representations of $G$ (up to equiv.) for $\oplus$ and $\otimes$. It is a fusion ring, and it is simple iff $G$ is simple. So the notion of simple fusion ring generalizes the notion of simple group; it does **not** correspond to the usual notion of simple ring.  

The fusion ring $\mathcal{G}_p$ is the Grothendieck ring of the cyclic group of prime order $p$.  

 We have checked by SAGE (by using [this code][1]) that the only integral simple fusion ring of Frobenius type, rank $\leq 5$ and FPdim $< 28470$ (except $\mathcal{G}_p$) is the Grothendieck ring of the simple group $A_5$. It is of rank $5$ and FPdim $60$.

**Question**: Is there an integral simple fusion ring of rank $ \leq 5$,  FPdim $>60$ and Frobenius type?    
___  
*Bonus for rank $6$:*  

 We have checked by SAGE that the only integral simple fusion ring of Frobenius type, rank $6$ and FPdim $< 3360$ (except $\mathcal{G}_p$) is the Grothendieck ring of the simple group ${\rm PSL}(2,7)$, of order $168$.

*Bonus question*: Is there an integral simple fusion ring of rank $6$,  FPdim $>168$ and Frobenius type? 
____
*Digression*: a fusion ring is called non-trivial if it is not the Grothendieck ring of a finite group. The first non-trivial integral simple fusion ring found by SAGE is of rank $7$ and FPdim $210$ (see [here][2]).


  [1]: https://drive.google.com/file/d/0B2P_JgZe-Zd0RG53WGdmZlFyRzQ/view?usp=sharing
  [2]: http://mathoverflow.net/q/132866/34538