\(\text{FR}^{9,2}_{45}\)
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{4} & \mathbf{3} & \mathbf{5} & \mathbf{7} & \mathbf{6} & \mathbf{9} & \mathbf{8} \\ \mathbf{3} & \mathbf{4} & \mathbf{1}+\mathbf{3}+\mathbf{5}+\mathbf{6}+\mathbf{7}+\mathbf{9} & \mathbf{2}+\mathbf{4}+\mathbf{5}+\mathbf{6}+\mathbf{7}+\mathbf{8} & \mathbf{3}+\mathbf{4}+\mathbf{5}+\mathbf{8}+\mathbf{9} & \mathbf{3}+\mathbf{4}+\mathbf{7}+\mathbf{8}+\mathbf{9} & \mathbf{3}+\mathbf{4}+\mathbf{6}+\mathbf{8}+\mathbf{9} & \mathbf{4}+\mathbf{5}+\mathbf{6}+\mathbf{7}+\mathbf{8}+\mathbf{9} & \mathbf{3}+\mathbf{5}+\mathbf{6}+\mathbf{7}+\mathbf{8}+\mathbf{9} \\ \mathbf{4} & \mathbf{3} & \mathbf{2}+\mathbf{4}+\mathbf{5}+\mathbf{6}+\mathbf{7}+\mathbf{8} & \mathbf{1}+\mathbf{3}+\mathbf{5}+\mathbf{6}+\mathbf{7}+\mathbf{9} & \mathbf{3}+\mathbf{4}+\mathbf{5}+\mathbf{8}+\mathbf{9} & \mathbf{3}+\mathbf{4}+\mathbf{6}+\mathbf{8}+\mathbf{9} & \mathbf{3}+\mathbf{4}+\mathbf{7}+\mathbf{8}+\mathbf{9} & \mathbf{3}+\mathbf{5}+\mathbf{6}+\mathbf{7}+\mathbf{8}+\mathbf{9} & \mathbf{4}+\mathbf{5}+\mathbf{6}+\mathbf{7}+\mathbf{8}+\mathbf{9} \\ \mathbf{5} & \mathbf{5} & \mathbf{3}+\mathbf{4}+\mathbf{5}+\mathbf{8}+\mathbf{9} & \mathbf{3}+\mathbf{4}+\mathbf{5}+\mathbf{8}+\mathbf{9} & \mathbf{1}+\mathbf{2}+\mathbf{3}+\mathbf{4}+\mathbf{5}+\mathbf{6}+\mathbf{7} & \mathbf{5}+\mathbf{6}+\mathbf{7}+\mathbf{8}+\mathbf{9} & \mathbf{5}+\mathbf{6}+\mathbf{7}+\mathbf{8}+\mathbf{9} & \mathbf{3}+\mathbf{4}+\mathbf{6}+\mathbf{7}+\mathbf{8}+\mathbf{9} & \mathbf{3}+\mathbf{4}+\mathbf{6}+\mathbf{7}+\mathbf{8}+\mathbf{9} \\ \mathbf{6} & \mathbf{7} & \mathbf{3}+\mathbf{4}+\mathbf{7}+\mathbf{8}+\mathbf{9} & \mathbf{3}+\mathbf{4}+\mathbf{6}+\mathbf{8}+\mathbf{9} & \mathbf{5}+\mathbf{6}+\mathbf{7}+\mathbf{8}+\mathbf{9} & \mathbf{2}+\mathbf{3}+\mathbf{5}+\mathbf{6}+\mathbf{7}+\mathbf{8} & \mathbf{1}+\mathbf{4}+\mathbf{5}+\mathbf{6}+\mathbf{7}+\mathbf{9} & \mathbf{3}+\mathbf{4}+\mathbf{5}+\mathbf{7}+\mathbf{8}+\mathbf{9} & \mathbf{3}+\mathbf{4}+\mathbf{5}+\mathbf{6}+\mathbf{8}+\mathbf{9} \\ \mathbf{7} & \mathbf{6} & \mathbf{3}+\mathbf{4}+\mathbf{6}+\mathbf{8}+\mathbf{9} & \mathbf{3}+\mathbf{4}+\mathbf{7}+\mathbf{8}+\mathbf{9} & \mathbf{5}+\mathbf{6}+\mathbf{7}+\mathbf{8}+\mathbf{9} & \mathbf{1}+\mathbf{4}+\mathbf{5}+\mathbf{6}+\mathbf{7}+\mathbf{9} & \mathbf{2}+\mathbf{3}+\mathbf{5}+\mathbf{6}+\mathbf{7}+\mathbf{8} & \mathbf{3}+\mathbf{4}+\mathbf{5}+\mathbf{6}+\mathbf{8}+\mathbf{9} & \mathbf{3}+\mathbf{4}+\mathbf{5}+\mathbf{7}+\mathbf{8}+\mathbf{9} \\ \mathbf{8} & \mathbf{9} & \mathbf{4}+\mathbf{5}+\mathbf{6}+\mathbf{7}+\mathbf{8}+\mathbf{9} & \mathbf{3}+\mathbf{5}+\mathbf{6}+\mathbf{7}+\mathbf{8}+\mathbf{9} & \mathbf{3}+\mathbf{4}+\mathbf{6}+\mathbf{7}+\mathbf{8}+\mathbf{9} & \mathbf{3}+\mathbf{4}+\mathbf{5}+\mathbf{7}+\mathbf{8}+\mathbf{9} & \mathbf{3}+\mathbf{4}+\mathbf{5}+\mathbf{6}+\mathbf{8}+\mathbf{9} & \mathbf{1}+\mathbf{3}+\mathbf{4}+\mathbf{5}+\mathbf{6}+\mathbf{7}+\mathbf{8}+\mathbf{9} & \mathbf{2}+\mathbf{3}+\mathbf{4}+\mathbf{5}+\mathbf{6}+\mathbf{7}+\mathbf{8}+\mathbf{9} \\ \mathbf{9} & \mathbf{8} & \mathbf{3}+\mathbf{5}+\mathbf{6}+\mathbf{7}+\mathbf{8}+\mathbf{9} & \mathbf{4}+\mathbf{5}+\mathbf{6}+\mathbf{7}+\mathbf{8}+\mathbf{9} & \mathbf{3}+\mathbf{4}+\mathbf{6}+\mathbf{7}+\mathbf{8}+\mathbf{9} & \mathbf{3}+\mathbf{4}+\mathbf{5}+\mathbf{6}+\mathbf{8}+\mathbf{9} & \mathbf{3}+\mathbf{4}+\mathbf{5}+\mathbf{7}+\mathbf{8}+\mathbf{9} & \mathbf{2}+\mathbf{3}+\mathbf{4}+\mathbf{5}+\mathbf{6}+\mathbf{7}+\mathbf{8}+\mathbf{9} & \mathbf{1}+\mathbf{3}+\mathbf{4}+\mathbf{5}+\mathbf{6}+\mathbf{7}+\mathbf{8}+\mathbf{9} \\ \hline \end{array}\]The fusion rules are invariant under the group generated by the following permutations:
\[\{(\mathbf{6} \ \mathbf{7})\}\]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 |
|---|---|---|
| \(\mathbf{1}\) | \(1.\) | \(1\) |
| \(\mathbf{2}\) | \(1.\) | \(1\) |
| \(\mathbf{3}\) | \(5.37228\) | \(\frac{1}{2} \left(5+\sqrt{33}\right)\) |
| \(\mathbf{4}\) | \(5.37228\) | \(\frac{1}{2} \left(5+\sqrt{33}\right)\) |
| \(\mathbf{5}\) | \(5.37228\) | \(\frac{1}{2} \left(5+\sqrt{33}\right)\) |
| \(\mathbf{6}\) | \(5.37228\) | \(\frac{1}{2} \left(5+\sqrt{33}\right)\) |
| \(\mathbf{7}\) | \(5.37228\) | \(\frac{1}{2} \left(5+\sqrt{33}\right)\) |
| \(\mathbf{8}\) | \(6.37228\) | \(\frac{1}{2} \left(7+\sqrt{33}\right)\) |
| \(\mathbf{9}\) | \(6.37228\) | \(\frac{1}{2} \left(7+\sqrt{33}\right)\) |
| \(\mathcal{D}_{FP}^2\) | \(227.519\) | \(2+\frac{5}{4} \left(5+\sqrt{33}\right)^2+\frac{1}{2} \left(7+\sqrt{33}\right)^2\) |
Characters
The symbolic character table is the following
\[\begin{array}{|ccccccccc|} \hline \mathbf{1} & \mathbf{2} & \mathbf{3} & \mathbf{5} & \mathbf{7} & \mathbf{6} & \mathbf{4} & \mathbf{8} & \mathbf{9} \\ \hline 1 & 1 & \frac{1}{2} \left(5+\sqrt{33}\right) & \frac{1}{2} \left(5+\sqrt{33}\right) & \frac{1}{2} \left(5+\sqrt{33}\right) & \frac{1}{2} \left(5+\sqrt{33}\right) & \frac{1}{2} \left(5+\sqrt{33}\right) & \frac{1}{2} \left(7+\sqrt{33}\right) & \frac{1}{2} \left(7+\sqrt{33}\right) \\ 1 & 1 & -2 & 2 & 2 & 2 & -2 & -1 & -1 \\ 1 & 1 & 1 & 2 & -1 & -1 & 1 & -1 & -1 \\ 1 & 1 & 1 & -2 & 1 & 1 & 1 & -1 & -1 \\ 1 & 1 & \frac{1}{2} \left(5-\sqrt{33}\right) & \frac{1}{2} \left(5-\sqrt{33}\right) & \frac{1}{2} \left(5-\sqrt{33}\right) & \frac{1}{2} \left(5-\sqrt{33}\right) & \frac{1}{2} \left(5-\sqrt{33}\right) & \frac{1}{2} \left(7-\sqrt{33}\right) & \frac{1}{2} \left(7-\sqrt{33}\right) \\ 1 & -1 & 0 & 0 & 0 & 0 & 0 & 1 & -1 \\ 1 & -1 & 2 & 0 & 0 & 0 & -2 & -1 & 1 \\ 1 & -1 & -1 & 0 & i \sqrt{3} & -i \sqrt{3} & 1 & -1 & 1 \\ 1 & -1 & -1 & 0 & -i \sqrt{3} & i \sqrt{3} & 1 & -1 & 1 \\ \hline \end{array}\]The numeric character table is the following
\[\begin{array}{|rrrrrrrrr|} \hline \mathbf{1} & \mathbf{2} & \mathbf{3} & \mathbf{5} & \mathbf{7} & \mathbf{6} & \mathbf{4} & \mathbf{8} & \mathbf{9} \\ \hline 1.000 & 1.000 & 5.372 & 5.372 & 5.372 & 5.372 & 5.372 & 6.372 & 6.372 \\ 1.000 & 1.000 & -2.000 & 2.000 & 2.000 & 2.000 & -2.000 & -1.000 & -1.000 \\ 1.000 & 1.000 & 1.000 & 2.000 & -1.000 & -1.000 & 1.000 & -1.000 & -1.000 \\ 1.000 & 1.000 & 1.000 & -2.000 & 1.000 & 1.000 & 1.000 & -1.000 & -1.000 \\ 1.000 & 1.000 & -0.3723 & -0.3723 & -0.3723 & -0.3723 & -0.3723 & 0.6277 & 0.6277 \\ 1.000 & -1.000 & 0 & 0 & 0 & 0 & 0 & 1.000 & -1.000 \\ 1.000 & -1.000 & 2.000 & 0 & 0 & 0 & -2.000 & -1.000 & 1.000 \\ 1.000 & -1.000 & -1.000 & 0 & 1.732 i & -1.732 i & 1.000 & -1.000 & 1.000 \\ 1.000 & -1.000 & -1.000 & 0 & -1.732 i & 1.732 i & 1.000 & -1.000 & 1.000 \\ \hline \end{array}\]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: