$\DeclareMathOperator\ch{ch}$$\DeclareMathOperator\FPdim{FPdim}$We refer to [EGNO15, Chapter 3] for the notion of fusion ring and basic results. The type of a fusion ring $R$ is the list $(\FPdim(b_i))$ where $(b_i)$ is its basis, i.e. the ($\ell^2$) operator norm of the fusion matrices. The $\FPdim$ of $R$ is $\sum_i \FPdim(b_i)^2$, its rank is the cardinal of its basis. It is called integral if $(\FPdim(b_i))$ are integers. The group ring $\mathbb{Z}G$ is an integral fusion ring, with basis the finite group $G$. Its $\FPdim$ and its rank equals $|G|$ (i.e. the pointed case). The character ring $\ch(G)$ is another integral fusion ring, with basis the set of irreducible characters of $G$. Its $\FPdim$ equals $|G|$, but its rank equals the class number of $G$. A fusion ring is called simple if it lacks any proper nontrivial fusion subrings. The fusion ring $\ch(G)$ is simple if and only if the group $G$ is itself simple. This fusion ring $\ch(G)$ remembers the simple group $G$; however, this is not the case for non-simple groups, such as $D_4$ and $Q_8$. Additionally, there are plenty of simple integral fusion rings that cannot be represented as $\ch(G)$. Therefore, a classification of simple integral fusion rings would really extend the CFSG. The smallest possible rank of a non-pointed simple integral fusion ring as $\ch(G)$ is $5$, corresponding $G=A_5$.
It has been established that a non-trivial perfect integral fusion ring must contain at least $4$ distinct basic FPdims, see [ABBP, Proposition 3.2]. Consequently, a non-pointed simple integral fusion ring must have a rank of at least $4$.
A simple integral fusion ring of rank $4$ must have all its basic elements self-dual, so by Frobenius reciprocity, the fusion matrices are generically of the form$$\left(\begin{matrix} 1&0&0&0 \\ 0&1&0&0 \\ 0&0&1&0 \\ 0&0&0&1\end{matrix} \right), \ \left(\begin{matrix} 0&1&0&0 \\ 1&x_1&x_2&x_3 \\ 0&x_2&x_4&x_5 \\ 0&x_3&x_5&x_6\end{matrix} \right), \ \left(\begin{matrix} 0&0&1&0 \\ 0&x_2&x_4&x_5 \\ 1&x_4&x_7&x_8 \\ 0&x_5&x_8&x_9\end{matrix} \right), \ \left(\begin{matrix} 0&0&0&1 \\ 0&x_3&x_5&x_6 \\ 0&x_5&x_8&x_9 \\ 1&x_6&x_9&x_{10} \end{matrix} \right),$$where $x_i$ are non-negative integers satisfying the following associativity equations:
$$x_{2}^2 - x_{1} x_{4} + x_{4}^2 + x_{5}^2 - x_{2} x_{7} - x_{3} x_{8} - 1 = 0, \\x_{5}^2 - x_{4} x_{6} - x_{10} x_{8} + x_{8}^2 - x_{7} x_{9} + x_{9}^2 - 1 = 0, \\x_{10} x_{3} - x_{3}^2 - x_{5}^2 + x_{1} x_{6} - x_{6}^2 + x_{2} x_{9} + 1 = 0, \\x_{3} x_{4} - x_{2} x_{5} + x_{5} x_{7} - x_{4} x_{8} + x_{6} x_{8} - x_{5} x_{9} = 0, \\x_{2} x_{3} - x_{1} x_{5} + x_{4} x_{5} + x_{5} x_{6} - x_{2} x_{8} - x_{3} x_{9} = 0, \\x_{10} x_{5} - x_{3} x_{5} + x_{2} x_{6} - x_{5} x_{8} + x_{4} x_{9} - x_{6} x_{9} = 0, $$such that the ($\ell^2$) operator norm of above matrices are integers $1, d_1, d_2, d_3$, and $3 \le d_1 < d_2 < d_3$. By Frobenius-Perron theorem, $\FPdim$ is a ring homomorphism, so we have the dimension equations:$$d_1^2 = 1 + d_1x_1 + d_2x_2 + d_3x_3, \\ d_1 d_2 = d_1x_2 + d_2x_4 + d_3x_5, \\ d_1 d_3 = d_1x_3 + d_2x_5 + d_3x_6, \\d_2^2 = 1 + d_1x_4 + d_2x_7 + d_3x_8, \\ d_2 d_3 = d_1x_5 + d_2x_8 + d_3x_9, \\d_3^2 = 1 + d_1x_6 + d_2x_9 + d_3x_{10}.$$For given integers $(d_i)$, the above (dimension and associativity) equations can be solved very efficiently by Normaliz, see [ABPP] and [BISV].
Up to isomorphism, there are $12$ simple integral fusion rings of rank $4$ and $\FPdim < 10^6$, see [BP, P1], presented in the table below (seemingly random yet number-theoretically intriguing):$$ \scriptsize\begin{array}{cc|ccc|cccccccccc}\mathrm{FPdim} & \text{Factors}& d_1 & d_2 & d_3 & x_1 & x_2 & x_3 & x_4 & x_5 & x_6 & x_7 & x_8 & x_9 & x_{10} \\ \hline574&2 \times 7 \times 41&11 & 14 & 16&0 & 4 & 4 & 1 & 6 & 3 & 12 & 1 & 9 & 6\\ \hline7315&5 \times 7 \times 11 \times 19& 35 & 40 & 67&30 & 1 & 2 & 9 & 15 & 25 & 12 & 12 & 25 & 39\\ \hline63436& 2^2 \times 1585 &103 & 149 & 175&57 & 13 & 16 & 67 & 23 & 74 & 51 & 44 & 98 & 48\\ \hline65971& 2 \times 3^2 \times 5 \times 1259 & 65 & 89 & 232&51 & 5 & 2 & 4 & 22 & 56 & 60 & 10 & 79 & 186\\ \hline68587& 107 \times 641 & 103 & 116 & 211&48 & 27 & 12 & 19 & 33 & 79 & 30 & 38 & 79 & 129\\ \hline90590& 2 \times 5 \times 9059 & 142 & 180 & 195&39 & 0 & 75 & 77 & 60 & 32 & 25 & 87 & 56 & 120\\ \hline113310& 2 \times 3^2 \times 5 \times 125 & 90 & 172 & 275&77 & 2 & 3 & 25 & 40 & 64 & 39 & 75 & 112 & 184\\ \hline310730& 2 \times 5 \times 7 \times 23 \times 193 & 312 & 317 & 336&62 & 139 & 101 & 48 & 120 & 105 & 168 & 96 & 115 & 130\\ \hline311343& 3 \times 59 \times 1759 & 286 & 315 & 361&95 & 76 & 85 & 115 & 89 & 141 & 206 & 4 & 241 & 39\\ \hline494102& 2 \times 7 \times 29 \times 1217 & 396 & 399 & 422&40 & 219 & 127 & 20 & 150 & 135 & 248 & 124 & 141 & 162\\ \hline532159& 532159 & 211 & 409 & 566&84 & 24 & 30 & 63 & 98 & 129 & 299 & 56 & 332 & 278\\ \hline585123& 3 \times 7 \times 11 \times 17 \times 149 & 288 & 397 & 587&159 & 30 & 43 & 179 & 59 & 227 & 208 & 40 & 341 & 245\end{array}$$
Question 1: Are there an infinite number of simple integral fusion rings of rank $4$?
The data in the aforementioned table not only hints at an affirmative response to Question 1 but also might aid in the discovery of an infinite family of such rings. The sequence of FPdims of such fusion rings has just been cataloged in OEIS, see [P2].
A fusion ring $R$ is called 1-Frobenius if the quotient $\FPdim(R)/\FPdim(b_i)$ is an algebraic integer for every basic element $b_i$. Thus, an integral fusion ring is $1$-Frobenius if $\FPdim(b_i)$ divides $\FPdim(R)$ for all $i$. The character ring $\ch(G)$ is $1$-Frobenius because every irreducible character degree divides $|G|$. Kaplansky's 6th conjecture generalized to fusion categories, as mentioned in [ENO11, Question 1], states that every fusion category (over $\mathbb{C}$) has $1$-Frobenius Grothendieck ring.
None of the fusion rings from the above table is $1$-Frobenius, more strongly, there are no $1$-Frobenius simple integral fusion rings of rank $4$ and $\FPdim < 10^{10}$, see [BP, P1].
Question 2: Is there a $1$-Frobenius simple integral fusion ring of rank $4$?
References:
[EGNO15] P. Etingof, S. Gelaki, D. Nikshych, and V. Ostrik, Tensor Categories, Mathematical Surveys and Monographs Volume 205 (2015).
[ENO11] P. Etingof, D. Nikshych, and V. Ostrik, Weakly group-theoretical and solvable fusion categories. Adv. Math. 226 (2011), no. 1, 176–205.
[ABPP] M.A. Alekseyev, W. Bruns, S. Palcoux, F.V. Petrov, Classification of modular data of integral modular fusion categories up to rank 12, arXiv:2302.01613v3
[BP] W. Bruns and S. Palcoux, Classification of simple integral fusion rings, work in progress.
[P1] S. Palcoux, Exotic Integral Quantum Symmetry, https://bimsa.net/doc/notes/23787.pdf
[P2] S. Palcoux, Frobenius-Perron dimensions of simple integral fusion rings of rank 4, https://oeis.org/A369625
[BISV] W. Bruns, B. Ichim, C. Söger and U. von der Ohe: Normaliz. Algorithms for rational cones and affine monoids. Available at https://www.normaliz.uni-osnabrueck.de.