\(\text{FR}^{9,4}_{24}\)
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{6} & \mathbf{5} & \mathbf{8} & \mathbf{7} & \mathbf{9} \\ \mathbf{3} & \mathbf{4} & \mathbf{2}+\mathbf{5} & \mathbf{1}+\mathbf{6} & \mathbf{4}+\mathbf{7} & \mathbf{3}+\mathbf{8} & \mathbf{6}+\mathbf{9} & \mathbf{5}+\mathbf{9} & \mathbf{7}+\mathbf{8} \\ \mathbf{4} & \mathbf{3} & \mathbf{1}+\mathbf{6} & \mathbf{2}+\mathbf{5} & \mathbf{3}+\mathbf{8} & \mathbf{4}+\mathbf{7} & \mathbf{5}+\mathbf{9} & \mathbf{6}+\mathbf{9} & \mathbf{7}+\mathbf{8} \\ \mathbf{5} & \mathbf{6} & \mathbf{4}+\mathbf{7} & \mathbf{3}+\mathbf{8} & \mathbf{1}+\mathbf{6}+\mathbf{9} & \mathbf{2}+\mathbf{5}+\mathbf{9} & \mathbf{3}+\mathbf{7}+\mathbf{8} & \mathbf{4}+\mathbf{7}+\mathbf{8} & \mathbf{5}+\mathbf{6}+\mathbf{9} \\ \mathbf{6} & \mathbf{5} & \mathbf{3}+\mathbf{8} & \mathbf{4}+\mathbf{7} & \mathbf{2}+\mathbf{5}+\mathbf{9} & \mathbf{1}+\mathbf{6}+\mathbf{9} & \mathbf{4}+\mathbf{7}+\mathbf{8} & \mathbf{3}+\mathbf{7}+\mathbf{8} & \mathbf{5}+\mathbf{6}+\mathbf{9} \\ \mathbf{7} & \mathbf{8} & \mathbf{6}+\mathbf{9} & \mathbf{5}+\mathbf{9} & \mathbf{3}+\mathbf{7}+\mathbf{8} & \mathbf{4}+\mathbf{7}+\mathbf{8} & \mathbf{2}+\mathbf{5}+\mathbf{6}+\mathbf{9} & \mathbf{1}+\mathbf{5}+\mathbf{6}+\mathbf{9} & \mathbf{3}+\mathbf{4}+\mathbf{7}+\mathbf{8} \\ \mathbf{8} & \mathbf{7} & \mathbf{5}+\mathbf{9} & \mathbf{6}+\mathbf{9} & \mathbf{4}+\mathbf{7}+\mathbf{8} & \mathbf{3}+\mathbf{7}+\mathbf{8} & \mathbf{1}+\mathbf{5}+\mathbf{6}+\mathbf{9} & \mathbf{2}+\mathbf{5}+\mathbf{6}+\mathbf{9} & \mathbf{3}+\mathbf{4}+\mathbf{7}+\mathbf{8} \\ \mathbf{9} & \mathbf{9} & \mathbf{7}+\mathbf{8} & \mathbf{7}+\mathbf{8} & \mathbf{5}+\mathbf{6}+\mathbf{9} & \mathbf{5}+\mathbf{6}+\mathbf{9} & \mathbf{3}+\mathbf{4}+\mathbf{7}+\mathbf{8} & \mathbf{3}+\mathbf{4}+\mathbf{7}+\mathbf{8} & \mathbf{1}+\mathbf{2}+\mathbf{5}+\mathbf{6}+\mathbf{9} \\ \hline \end{array}\]The fusion rules are invariant under the group generated by the following permutations:
\[\{(\mathbf{3} \ \mathbf{4}) (\mathbf{7} \ \mathbf{8})\}\]The following particles form non-trivial sub fusion rings
| Particles | SubRing |
|---|---|
| \(\{\mathbf{1},\mathbf{2}\}\) | \(\mathbb{Z}_2:\ \text{FR}^{2,0}_{1}\) |
| \(\{\mathbf{1},\mathbf{2},\mathbf{5},\mathbf{6},\mathbf{9}\}\) | \(\text{PSU(2})_8:\ \text{FR}^{5,0}_{7}\) |
Quantum Dimensions
| Particle | Numeric | Symbolic |
|---|---|---|
| \(\mathbf{1}\) | \(1.\) | \(1\) |
| \(\mathbf{2}\) | \(1.\) | \(1\) |
| \(\mathbf{3}\) | \(1.90211\) | \(\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}\) |
| \(\mathbf{4}\) | \(1.90211\) | \(\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}\) |
| \(\mathbf{5}\) | \(2.61803\) | \(\frac{1}{2} \left(3+\sqrt{5}\right)\) |
| \(\mathbf{6}\) | \(2.61803\) | \(\frac{1}{2} \left(3+\sqrt{5}\right)\) |
| \(\mathbf{7}\) | \(3.07768\) | \(\sqrt{5+2 \sqrt{5}}\) |
| \(\mathbf{8}\) | \(3.07768\) | \(\sqrt{5+2 \sqrt{5}}\) |
| \(\mathbf{9}\) | \(3.23607\) | \(1+\sqrt{5}\) |
| \(\mathcal{D}_{FP}^2\) | \(52.3607\) | \(17+5 \sqrt{5}+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2\) |
Characters
The symbolic character table is the following
\[\begin{array}{|ccccccccc|} \hline \mathbf{1} & \mathbf{2} & \mathbf{3} & \mathbf{4} & \mathbf{5} & \mathbf{6} & \mathbf{8} & \mathbf{7} & \mathbf{9} \\ \hline 1. & 1. & 1.90211 & 1.90211 & 2.61803 & 2.61803 & 3.07768 & 3.07768 & 3.23607 \\ 1. & 1. & 1.17557 & 1.17557 & 0.381966 & 0.381966 & -0.726543 & -0.726543 & -1.23607 \\ 1. & 1. & 0. & 0. & -1. & -1. & 0. & 0. & 1. \\ 1. & 1. & -1.17557 & -1.17557 & 0.381966 & 0.381966 & 0.726543 & 0.726543 & -1.23607 \\ 1. & 1. & -1.90211 & -1.90211 & 2.61803 & 2.61803 & -3.07768 & -3.07768 & 3.23607 \\ 1. & -1.+0. i & 0.618034 i & 0.\, -0.618034 i & 0.618034\, +0. i & -0.618034+0. i & 0.\, -1. i & 0.\, +1. i & 0. \\ 1. & -1.+0. i & 0.\, -0.618034 i & 0.\, +0.618034 i & 0.618034\, +0. i & -0.618034+0. i & 0.\, +1. i & 0.\, -1. i & 0. \\ 1.\, +0. i & -1.+0. i & 0.\, +1.61803 i & 0.\, -1.61803 i & -1.61803+0. i & 1.61803\, +0. i & 0.\, +1. i & 0.\, -1. i & 0. \\ 1.\, +0. i & -1.+0. i & 0.\, -1.61803 i & 0.\, +1.61803 i & -1.61803+0. i & 1.61803\, +0. i & 0.\, -1. i & 0.\, +1. i & 0. \\ \hline \end{array}\]The numeric character table is the following
\[\begin{array}{|rrrrrrrrr|} \hline \mathbf{1} & \mathbf{2} & \mathbf{3} & \mathbf{4} & \mathbf{5} & \mathbf{6} & \mathbf{8} & \mathbf{7} & \mathbf{9} \\ \hline 1. & 1. & 1.90211 & 1.90211 & 2.61803 & 2.61803 & 3.07768 & 3.07768 & 3.23607 \\ 1. & 1. & 1.17557 & 1.17557 & 0.381966 & 0.381966 & -0.726543 & -0.726543 & -1.23607 \\ 1. & 1. & 0. & 0. & -1. & -1. & 0. & 0. & 1. \\ 1. & 1. & -1.17557 & -1.17557 & 0.381966 & 0.381966 & 0.726543 & 0.726543 & -1.23607 \\ 1. & 1. & -1.90211 & -1.90211 & 2.61803 & 2.61803 & -3.07768 & -3.07768 & 3.23607 \\ 1. & -1.+0. i & 0.\, +0.618034 i & 0.\, -0.618034 i & 0.618034\, +0. i & -0.618034+0. i & 0.\, -1. i & 0.\, +1. i & 0. \\ 1. & -1.+0. i & 0.\, -0.618034 i & 0.\, +0.618034 i & 0.618034\, +0. i & -0.618034+0. i & 0.\, +1. i & 0.\, -1. i & 0. \\ 1.\, +0. i & -1.+0. i & 0.\, +1.61803 i & 0.\, -1.61803 i & -1.61803+0. i & 1.61803\, +0. i & 0.\, +1. i & 0.\, -1. i & 0. \\ 1.\, +0. i & -1.+0. i & 0.\, -1.61803 i & 0.\, +1.61803 i & -1.61803+0. i & 1.61803\, +0. i & 0.\, -1. i & 0.\, +1. i & 0. \\ \hline \end{array}\]Modular Data
This fusion ring does not have any matching \(S\)-and \(T\)-matrices.
Adjoint Subring
Particles \(\mathbf{1}, \mathbf{2}, \mathbf{5}, \mathbf{6}, \mathbf{9}\), form the adjoint subring \(\text{PSU(2})_8:\ \text{FR}^{5,0}_{7}\) .
The upper central series is the following: \(\text{FR}^{9,4}_{24} \underset{ \mathbf{1}, \mathbf{2}, \mathbf{5}, \mathbf{6}, \mathbf{9} }{\supset} \text{PSU(2})_8\)
Universal grading
Each particle can be graded as follows: \(\text{deg}(\mathbf{1}) = \mathbf{1}', \text{deg}(\mathbf{2}) = \mathbf{1}', \text{deg}(\mathbf{3}) = \mathbf{2}', \text{deg}(\mathbf{4}) = \mathbf{2}', \text{deg}(\mathbf{5}) = \mathbf{1}', \text{deg}(\mathbf{6}) = \mathbf{1}', \text{deg}(\mathbf{7}) = \mathbf{2}', \text{deg}(\mathbf{8}) = \mathbf{2}', \text{deg}(\mathbf{9}) = \mathbf{1}'\), where the degrees form the group \(\mathbb{Z}_2\) with multiplication table:
\[\begin{array}{|ll|} \hline \mathbf{1}' & \mathbf{2}' \\ \mathbf{2}' & \mathbf{1}' \\ \hline \end{array}\]Categorifications
Data
Download links for numeric data: