\(\text{FR}^{9,6}_{8}\)
Fusion Rules
\[\begin{array}{|lllllllll|} \hline \mathbf{1} & \mathbf{2} & \mathbf{3} & \mathbf{4} & \mathbf{5} & \mathbf{6} & \mathbf{7} & \mathbf{8} & \mathbf{9} \\ \mathbf{2} & \mathbf{1} & \mathbf{6} & \mathbf{5} & \mathbf{4} & \mathbf{3} & \mathbf{7} & \mathbf{8} & \mathbf{9} \\ \mathbf{3} & \mathbf{6} & \mathbf{4} & \mathbf{1} & \mathbf{2} & \mathbf{5} & \mathbf{8} & \mathbf{9} & \mathbf{7} \\ \mathbf{4} & \mathbf{5} & \mathbf{1} & \mathbf{3} & \mathbf{6} & \mathbf{2} & \mathbf{9} & \mathbf{7} & \mathbf{8} \\ \mathbf{5} & \mathbf{4} & \mathbf{2} & \mathbf{6} & \mathbf{3} & \mathbf{1} & \mathbf{9} & \mathbf{7} & \mathbf{8} \\ \mathbf{6} & \mathbf{3} & \mathbf{5} & \mathbf{2} & \mathbf{1} & \mathbf{4} & \mathbf{8} & \mathbf{9} & \mathbf{7} \\ \mathbf{7} & \mathbf{7} & \mathbf{8} & \mathbf{9} & \mathbf{9} & \mathbf{8} & \mathbf{1}+\mathbf{2}+\mathbf{8}+\mathbf{9} & \mathbf{3}+\mathbf{6}+\mathbf{7}+\mathbf{9} & \mathbf{4}+\mathbf{5}+\mathbf{7}+\mathbf{8} \\ \mathbf{8} & \mathbf{8} & \mathbf{9} & \mathbf{7} & \mathbf{7} & \mathbf{9} & \mathbf{3}+\mathbf{6}+\mathbf{7}+\mathbf{9} & \mathbf{4}+\mathbf{5}+\mathbf{7}+\mathbf{8} & \mathbf{1}+\mathbf{2}+\mathbf{8}+\mathbf{9} \\ \mathbf{9} & \mathbf{9} & \mathbf{7} & \mathbf{8} & \mathbf{8} & \mathbf{7} & \mathbf{4}+\mathbf{5}+\mathbf{7}+\mathbf{8} & \mathbf{1}+\mathbf{2}+\mathbf{8}+\mathbf{9} & \mathbf{3}+\mathbf{6}+\mathbf{7}+\mathbf{9} \\ \hline \end{array}\]The fusion rules are invariant under the group generated by the following permutations:
\[\{(\mathbf{3} \ \mathbf{4}) (\mathbf{5} \ \mathbf{6}) (\mathbf{8} \ \mathbf{9})\}\]The following particles form non-trivial sub fusion rings
| Particles | SubRing |
|---|---|
| \(\{\mathbf{1},\mathbf{2}\}\) | \(\mathbb{Z}_2:\ \text{FR}^{2,0}_{1}\) |
| \(\{\mathbf{1},\mathbf{3},\mathbf{4}\}\) | \(\mathbb{Z}_3:\ \text{FR}^{3,2}_{1}\) |
| \(\{\mathbf{1},\mathbf{2},\mathbf{3},\mathbf{4},\mathbf{5},\mathbf{6}\}\) | \(\mathbb{Z}_6:\ \text{FR}^{6,4}_{1}\) |
Quantum Dimensions
| Particle | Numeric | Symbolic |
|---|---|---|
| \(\mathbf{1}\) | \(1.\) | \(1\) |
| \(\mathbf{2}\) | \(1.\) | \(1\) |
| \(\mathbf{3}\) | \(1.\) | \(1\) |
| \(\mathbf{4}\) | \(1.\) | \(1\) |
| \(\mathbf{5}\) | \(1.\) | \(1\) |
| \(\mathbf{6}\) | \(1.\) | \(1\) |
| \(\mathbf{7}\) | \(2.73205\) | \(1+\sqrt{3}\) |
| \(\mathbf{8}\) | \(2.73205\) | \(1+\sqrt{3}\) |
| \(\mathbf{9}\) | \(2.73205\) | \(1+\sqrt{3}\) |
| \(\mathcal{D}_{FP}^2\) | \(28.3923\) | \(6+3 \left(1+\sqrt{3}\right)^2\) |
Characters
The symbolic character table is the following
\[\begin{array}{|ccccccccc|} \hline \mathbf{1} & \mathbf{4} & \mathbf{5} & \mathbf{3} & \mathbf{6} & \mathbf{2} & \mathbf{7} & \mathbf{8} & \mathbf{9} \\ \hline 1 & 1 & 1 & 1 & 1 & 1 & 1+\sqrt{3} & 1+\sqrt{3} & 1+\sqrt{3} \\ 1 & 1 & 1 & 1 & 1 & 1 & 1-\sqrt{3} & 1-\sqrt{3} & 1-\sqrt{3} \\ 1 & 1 & -1 & 1 & -1 & -1 & 0 & 0 & 0 \\ 1 & \frac{1}{2} \left(-1-i \sqrt{3}\right) & \frac{1}{2} \left(1+i \sqrt{3}\right) & \frac{1}{2} \left(-1+i \sqrt{3}\right) & \frac{1}{2} \left(1-i \sqrt{3}\right) & -1 & 0 & 0 & 0 \\ 1 & \frac{1}{2} \left(-1+i \sqrt{3}\right) & \frac{1}{2} \left(1-i \sqrt{3}\right) & \frac{1}{2} \left(-1-i \sqrt{3}\right) & \frac{1}{2} \left(1+i \sqrt{3}\right) & -1 & 0 & 0 & 0 \\ 1 & \frac{1}{2} \left(-1+i \sqrt{3}\right) & \frac{1}{2} \left(-1+i \sqrt{3}\right) & \frac{1}{2} \left(-1-i \sqrt{3}\right) & \frac{1}{2} \left(-1-i \sqrt{3}\right) & 1 & 1 & \frac{1}{2} \left(-1-i \sqrt{3}\right) & \frac{1}{2} \left(-1+i \sqrt{3}\right) \\ 1 & \frac{1}{2} \left(-1-i \sqrt{3}\right) & \frac{1}{2} \left(-1-i \sqrt{3}\right) & \frac{1}{2} \left(-1+i \sqrt{3}\right) & \frac{1}{2} \left(-1+i \sqrt{3}\right) & 1 & 1 & \frac{1}{2} \left(-1+i \sqrt{3}\right) & \frac{1}{2} \left(-1-i \sqrt{3}\right) \\ 1 & \frac{1}{2} \left(-1+i \sqrt{3}\right) & \frac{1}{2} \left(-1+i \sqrt{3}\right) & \frac{1}{2} \left(-1-i \sqrt{3}\right) & \frac{1}{2} \left(-1-i \sqrt{3}\right) & 1 & -2 & 1+i \sqrt{3} & 1-i \sqrt{3} \\ 1 & \frac{1}{2} \left(-1-i \sqrt{3}\right) & \frac{1}{2} \left(-1-i \sqrt{3}\right) & \frac{1}{2} \left(-1+i \sqrt{3}\right) & \frac{1}{2} \left(-1+i \sqrt{3}\right) & 1 & -2 & 1-i \sqrt{3} & 1+i \sqrt{3} \\ \hline \end{array}\]The numeric character table is the following
\[\begin{array}{|rrrrrrrrr|} \hline \mathbf{1} & \mathbf{4} & \mathbf{5} & \mathbf{3} & \mathbf{6} & \mathbf{2} & \mathbf{7} & \mathbf{8} & \mathbf{9} \\ \hline 1.000 & 1.000 & 1.000 & 1.000 & 1.000 & 1.000 & 2.732 & 2.732 & 2.732 \\ 1.000 & 1.000 & 1.000 & 1.000 & 1.000 & 1.000 & -0.7321 & -0.7321 & -0.7321 \\ 1.000 & 1.000 & -1.000 & 1.000 & -1.000 & -1.000 & 0 & 0 & 0 \\ 1.000 & -0.5000-0.8660 i & 0.5000+0.8660 i & -0.5000+0.8660 i & 0.5000-0.8660 i & -1.000 & 0 & 0 & 0 \\ 1.000 & -0.5000+0.8660 i & 0.5000-0.8660 i & -0.5000-0.8660 i & 0.5000+0.8660 i & -1.000 & 0 & 0 & 0 \\ 1.000 & -0.5000+0.8660 i & -0.5000+0.8660 i & -0.5000-0.8660 i & -0.5000-0.8660 i & 1.000 & 1.000 & -0.5000-0.8660 i & -0.5000+0.8660 i \\ 1.000 & -0.5000-0.8660 i & -0.5000-0.8660 i & -0.5000+0.8660 i & -0.5000+0.8660 i & 1.000 & 1.000 & -0.5000+0.8660 i & -0.5000-0.8660 i \\ 1.000 & -0.5000+0.8660 i & -0.5000+0.8660 i & -0.5000-0.8660 i & -0.5000-0.8660 i & 1.000 & -2.000 & 1.000+1.732 i & 1.000-1.732 i \\ 1.000 & -0.5000-0.8660 i & -0.5000-0.8660 i & -0.5000+0.8660 i & -0.5000+0.8660 i & 1.000 & -2.000 & 1.000-1.732 i & 1.000+1.732 i \\ \hline \end{array}\]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 it is quadratic (i.e. its fusion categories are pivotal) and does not satisfy the pivotal version of the Drinfeld center criterion.
Data
Download links for numeric data: