\(\text{FR}^{8,0}_{35}\)
Fusion Rules
\[\begin{array}{|llllllll|} \hline \mathbf{1} & \mathbf{2} & \mathbf{3} & \mathbf{4} & \mathbf{5} & \mathbf{6} & \mathbf{7} & \mathbf{8} \\ \mathbf{2} & \mathbf{1} & \mathbf{4} & \mathbf{3} & \mathbf{8} & \mathbf{7} & \mathbf{6} & \mathbf{5} \\ \mathbf{3} & \mathbf{4} & \mathbf{1}+\mathbf{4}+\mathbf{7}+\mathbf{8} & \mathbf{2}+\mathbf{3}+\mathbf{5}+\mathbf{6} & \mathbf{4}+\mathbf{5}+\mathbf{7}+\mathbf{8} & \mathbf{4}+\mathbf{6}+\mathbf{7}+\mathbf{8} & \mathbf{3}+\mathbf{5}+\mathbf{6}+\mathbf{7} & \mathbf{3}+\mathbf{5}+\mathbf{6}+\mathbf{8} \\ \mathbf{4} & \mathbf{3} & \mathbf{2}+\mathbf{3}+\mathbf{5}+\mathbf{6} & \mathbf{1}+\mathbf{4}+\mathbf{7}+\mathbf{8} & \mathbf{3}+\mathbf{5}+\mathbf{6}+\mathbf{8} & \mathbf{3}+\mathbf{5}+\mathbf{6}+\mathbf{7} & \mathbf{4}+\mathbf{6}+\mathbf{7}+\mathbf{8} & \mathbf{4}+\mathbf{5}+\mathbf{7}+\mathbf{8} \\ \mathbf{5} & \mathbf{8} & \mathbf{4}+\mathbf{5}+\mathbf{7}+\mathbf{8} & \mathbf{3}+\mathbf{5}+\mathbf{6}+\mathbf{8} & \mathbf{1}+\mathbf{3}+\mathbf{4}+\mathbf{6}+\mathbf{7}+\mathbf{8} & \mathbf{4}+\mathbf{5}+\mathbf{6}+\mathbf{7}+\mathbf{8} & \mathbf{3}+\mathbf{5}+\mathbf{6}+\mathbf{7}+\mathbf{8} & \mathbf{2}+\mathbf{3}+\mathbf{4}+\mathbf{5}+\mathbf{6}+\mathbf{7} \\ \mathbf{6} & \mathbf{7} & \mathbf{4}+\mathbf{6}+\mathbf{7}+\mathbf{8} & \mathbf{3}+\mathbf{5}+\mathbf{6}+\mathbf{7} & \mathbf{4}+\mathbf{5}+\mathbf{6}+\mathbf{7}+\mathbf{8} & \mathbf{1}+\mathbf{3}+\mathbf{4}+\mathbf{5}+\mathbf{7}+\mathbf{8} & \mathbf{2}+\mathbf{3}+\mathbf{4}+\mathbf{5}+\mathbf{6}+\mathbf{8} & \mathbf{3}+\mathbf{5}+\mathbf{6}+\mathbf{7}+\mathbf{8} \\ \mathbf{7} & \mathbf{6} & \mathbf{3}+\mathbf{5}+\mathbf{6}+\mathbf{7} & \mathbf{4}+\mathbf{6}+\mathbf{7}+\mathbf{8} & \mathbf{3}+\mathbf{5}+\mathbf{6}+\mathbf{7}+\mathbf{8} & \mathbf{2}+\mathbf{3}+\mathbf{4}+\mathbf{5}+\mathbf{6}+\mathbf{8} & \mathbf{1}+\mathbf{3}+\mathbf{4}+\mathbf{5}+\mathbf{7}+\mathbf{8} & \mathbf{4}+\mathbf{5}+\mathbf{6}+\mathbf{7}+\mathbf{8} \\ \mathbf{8} & \mathbf{5} & \mathbf{3}+\mathbf{5}+\mathbf{6}+\mathbf{8} & \mathbf{4}+\mathbf{5}+\mathbf{7}+\mathbf{8} & \mathbf{2}+\mathbf{3}+\mathbf{4}+\mathbf{5}+\mathbf{6}+\mathbf{7} & \mathbf{3}+\mathbf{5}+\mathbf{6}+\mathbf{7}+\mathbf{8} & \mathbf{4}+\mathbf{5}+\mathbf{6}+\mathbf{7}+\mathbf{8} & \mathbf{1}+\mathbf{3}+\mathbf{4}+\mathbf{6}+\mathbf{7}+\mathbf{8} \\ \hline \end{array}\]The fusion rules are invariant under the group generated by the following permutations:
\[\{(\mathbf{5} \ \mathbf{6}) (\mathbf{7} \ \mathbf{8})\}\]The following particles form non-trivial sub fusion rings
| Particles | SubRing |
|---|---|
| \(\{\mathbf{1},\mathbf{2}\}\) | \(\mathbb{Z}_2:\ \text{FR}^{2,0}_{1}\) |
Quantum Dimensions
| Particle | Numeric | Symbolic |
|---|---|---|
| \(\mathbf{1}\) | \(1.\) | \(1\) |
| \(\mathbf{2}\) | \(1.\) | \(1\) |
| \(\mathbf{3}\) | \(3.79129\) | \(\frac{1}{2} \left(3+\sqrt{21}\right)\) |
| \(\mathbf{4}\) | \(3.79129\) | \(\frac{1}{2} \left(3+\sqrt{21}\right)\) |
| \(\mathbf{5}\) | \(4.79129\) | \(\frac{1}{2} \left(5+\sqrt{21}\right)\) |
| \(\mathbf{6}\) | \(4.79129\) | \(\frac{1}{2} \left(5+\sqrt{21}\right)\) |
| \(\mathbf{7}\) | \(4.79129\) | \(\frac{1}{2} \left(5+\sqrt{21}\right)\) |
| \(\mathbf{8}\) | \(4.79129\) | \(\frac{1}{2} \left(5+\sqrt{21}\right)\) |
| \(\mathcal{D}_{FP}^2\) | \(122.573\) | \(2+\frac{1}{2} \left(3+\sqrt{21}\right)^2+\left(5+\sqrt{21}\right)^2\) |
Characters
The symbolic character table is the following
\[\begin{array}{|cccccccc|} \hline \mathbf{1} & \mathbf{2} & \mathbf{4} & \mathbf{3} & \mathbf{5} & \mathbf{7} & \mathbf{6} & \mathbf{8} \\ \hline 1 & 1 & \frac{1}{2} \left(3+\sqrt{21}\right) & \frac{1}{2} \left(3+\sqrt{21}\right) & \frac{1}{2} \left(5+\sqrt{21}\right) & \frac{1}{2} \left(5+\sqrt{21}\right) & \frac{1}{2} \left(5+\sqrt{21}\right) & \frac{1}{2} \left(5+\sqrt{21}\right) \\ 1 & 1 & 1 & 1 & \frac{1}{2} \left(\sqrt{5}-1\right) & \frac{1}{2} \left(-1-\sqrt{5}\right) & \frac{1}{2} \left(-1-\sqrt{5}\right) & \frac{1}{2} \left(\sqrt{5}-1\right) \\ 1 & 1 & 1 & 1 & \frac{1}{2} \left(-1-\sqrt{5}\right) & \frac{1}{2} \left(\sqrt{5}-1\right) & \frac{1}{2} \left(\sqrt{5}-1\right) & \frac{1}{2} \left(-1-\sqrt{5}\right) \\ 1 & 1 & \frac{1}{2} \left(3-\sqrt{21}\right) & \frac{1}{2} \left(3-\sqrt{21}\right) & \frac{1}{2} \left(5-\sqrt{21}\right) & \frac{1}{2} \left(5-\sqrt{21}\right) & \frac{1}{2} \left(5-\sqrt{21}\right) & \frac{1}{2} \left(5-\sqrt{21}\right) \\ 1 & -1 & -1 & 1 & \frac{1}{2} \left(\sqrt{5}-1\right) & \frac{1}{2} \left(1+\sqrt{5}\right) & \frac{1}{2} \left(-1-\sqrt{5}\right) & \frac{1}{2} \left(1-\sqrt{5}\right) \\ 1 & -1 & \frac{1}{2} \left(3+\sqrt{5}\right) & \frac{1}{2} \left(-3-\sqrt{5}\right) & \frac{1}{2} \left(-1-\sqrt{5}\right) & \frac{1}{2} \left(1+\sqrt{5}\right) & \frac{1}{2} \left(-1-\sqrt{5}\right) & \frac{1}{2} \left(1+\sqrt{5}\right) \\ 1 & -1 & \frac{1}{2} \left(3-\sqrt{5}\right) & \frac{1}{2} \left(\sqrt{5}-3\right) & \frac{1}{2} \left(\sqrt{5}-1\right) & \frac{1}{2} \left(1-\sqrt{5}\right) & \frac{1}{2} \left(\sqrt{5}-1\right) & \frac{1}{2} \left(1-\sqrt{5}\right) \\ 1 & -1 & -1 & 1 & \frac{1}{2} \left(-1-\sqrt{5}\right) & \frac{1}{2} \left(1-\sqrt{5}\right) & \frac{1}{2} \left(\sqrt{5}-1\right) & \frac{1}{2} \left(1+\sqrt{5}\right) \\ \hline \end{array}\]The numeric character table is the following
\[\begin{array}{|rrrrrrrr|} \hline \mathbf{1} & \mathbf{2} & \mathbf{4} & \mathbf{3} & \mathbf{5} & \mathbf{7} & \mathbf{6} & \mathbf{8} \\ \hline 1.000 & 1.000 & 3.791 & 3.791 & 4.791 & 4.791 & 4.791 & 4.791 \\ 1.000 & 1.000 & 1.000 & 1.000 & 0.6180 & -1.618 & -1.618 & 0.6180 \\ 1.000 & 1.000 & 1.000 & 1.000 & -1.618 & 0.6180 & 0.6180 & -1.618 \\ 1.000 & 1.000 & -0.7913 & -0.7913 & 0.2087 & 0.2087 & 0.2087 & 0.2087 \\ 1.000 & -1.000 & -1.000 & 1.000 & 0.6180 & 1.618 & -1.618 & -0.6180 \\ 1.000 & -1.000 & 2.618 & -2.618 & -1.618 & 1.618 & -1.618 & 1.618 \\ 1.000 & -1.000 & 0.3820 & -0.3820 & 0.6180 & -0.6180 & 0.6180 & -0.6180 \\ 1.000 & -1.000 & -1.000 & 1.000 & -1.618 & -0.6180 & 0.6180 & 1.618 \\ \hline \end{array}\]Modular Data
This fusion ring does not have any matching \(S\)-and \(T\)-matrices.
Adjoint Subring
The adjoint subring is the ring itself.
The upper central series is trivial.
Universal grading
This fusion ring allows only the trivial grading.
Categorifications
This fusion ring has no categorifications because of the $d$-number criterion.
Data
Download links for numeric data: