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) that the only integral simple fusion ring of Frobenius type, rank $\leq 5$ and FPdim $< 30750$ (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 $< 3564$ 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?

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).

For $210 <$ FPdim $<1080$ and rank $7$, there are two integral simple fusion rings, both of FPdim $360$, one is the Grothendieck ring of the simple group $A_6$, the other is non-trivial (see the first two here).