A fusion ring is a finite dimensional $\mathbb{Z}$-module $\mathbb{Z}\mathcal{B}$ 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: there is an involution $i \mapsto i^*$ such that $n_{i^*j}^1=\delta_{i,j}$
- Frobenius-Perron reciprocity: $n_{ij}^k = n_{i^*k}^j = n_{kj^*}^i$,
- Associativity: $\sum_s n_{ij}^sn_{sk}^t = \sum_s n_{jk}^sn_{is}^t$
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 $\frac{FPdim(\mathbb{Z}\mathcal{B})}{d(h_i)}$ is an algebraic integer.
It is simple if $r>1$ and for any fusion subring $\mathbb{Z}\mathcal{S} \subseteq \mathbb{Z}\mathcal{B}$ with $\mathcal{S} \subseteq \mathcal{B}$, then $\mathcal{S} = \{ h_1 \}$ or $\mathcal{B}$.
Open problem: Every Grothendieck ring of a complex fusion category is of Frobenius type.
Remark: The Grothendieck ring of a finite group $Rep(G)$ 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 fusion ring based on the cyclic group of prime order $p$.
We checked by SAGE (by using this code) that the only integral simple fusion ring of Frobenius type, rank $\leq 5$ and FPdim $< 1000000$ (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 $< 150000$ 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-group-like if it is not the Grothendieck ring of $Rep(G)$ or $Vec(G)$ with $G$ a finite group. The first non-group-like integral simple fusion ring found by SAGE is of rank $7$ and FPdim $210$ (see here).
For $210 <$ FPdim $<15000$ and rank $7$, there are six integral simple fusion rings, two of FPdim $360$, one of which is the Grothendieck ring of the simple group $A_6$, the other is non-trivial, and four of FPdim $7980$ and type $[[1, 1], [19, 1], [20, 1], [21, 1], [42, 2], [57, 1]]$. See this paper.
