\(\text{Ising$\times $}\text{PSU}(2)_5:\ \text{FR}^{9,0}_{14}\)
Fusion Rules
\[\begin{array}{|lllllllll|} \hline \mathbf{1} & \mathbf{2} & \mathbf{3} & \mathbf{4} & \mathbf{5} & \mathbf{6} & \mathbf{7} & \mathbf{8} & \mathbf{9} \\ \mathbf{2} & \mathbf{1} & \mathbf{3} & \mathbf{5} & \mathbf{4} & \mathbf{7} & \mathbf{6} & \mathbf{8} & \mathbf{9} \\ \mathbf{3} & \mathbf{3} & \mathbf{1}+\mathbf{2} & \mathbf{8} & \mathbf{8} & \mathbf{9} & \mathbf{9} & \mathbf{4}+\mathbf{5} & \mathbf{6}+\mathbf{7} \\ \mathbf{4} & \mathbf{5} & \mathbf{8} & \mathbf{1}+\mathbf{6} & \mathbf{2}+\mathbf{7} & \mathbf{4}+\mathbf{6} & \mathbf{5}+\mathbf{7} & \mathbf{3}+\mathbf{9} & \mathbf{8}+\mathbf{9} \\ \mathbf{5} & \mathbf{4} & \mathbf{8} & \mathbf{2}+\mathbf{7} & \mathbf{1}+\mathbf{6} & \mathbf{5}+\mathbf{7} & \mathbf{4}+\mathbf{6} & \mathbf{3}+\mathbf{9} & \mathbf{8}+\mathbf{9} \\ \mathbf{6} & \mathbf{7} & \mathbf{9} & \mathbf{4}+\mathbf{6} & \mathbf{5}+\mathbf{7} & \mathbf{1}+\mathbf{4}+\mathbf{6} & \mathbf{2}+\mathbf{5}+\mathbf{7} & \mathbf{8}+\mathbf{9} & \mathbf{3}+\mathbf{8}+\mathbf{9} \\ \mathbf{7} & \mathbf{6} & \mathbf{9} & \mathbf{5}+\mathbf{7} & \mathbf{4}+\mathbf{6} & \mathbf{2}+\mathbf{5}+\mathbf{7} & \mathbf{1}+\mathbf{4}+\mathbf{6} & \mathbf{8}+\mathbf{9} & \mathbf{3}+\mathbf{8}+\mathbf{9} \\ \mathbf{8} & \mathbf{8} & \mathbf{4}+\mathbf{5} & \mathbf{3}+\mathbf{9} & \mathbf{3}+\mathbf{9} & \mathbf{8}+\mathbf{9} & \mathbf{8}+\mathbf{9} & \mathbf{1}+\mathbf{2}+\mathbf{6}+\mathbf{7} & \mathbf{4}+\mathbf{5}+\mathbf{6}+\mathbf{7} \\ \mathbf{9} & \mathbf{9} & \mathbf{6}+\mathbf{7} & \mathbf{8}+\mathbf{9} & \mathbf{8}+\mathbf{9} & \mathbf{3}+\mathbf{8}+\mathbf{9} & \mathbf{3}+\mathbf{8}+\mathbf{9} & \mathbf{4}+\mathbf{5}+\mathbf{6}+\mathbf{7} & \mathbf{1}+\mathbf{2}+\mathbf{4}+\mathbf{5}+\mathbf{6}+\mathbf{7} \\ \hline \end{array}\]The following particles form non-trivial sub fusion rings
| Particles | SubRing |
|---|---|
| \(\{\mathbf{1},\mathbf{2}\}\) | \(\mathbb{Z}_2:\ \text{FR}^{2,0}_{1}\) |
| \(\{\mathbf{1},\mathbf{2},\mathbf{3}\}\) | \(\text{Ising}:\ \text{FR}^{3,0}_{1}\) |
| \(\{\mathbf{1},\mathbf{4},\mathbf{6}\}\) | \(\text{PSU}(2)_5:\ \text{FR}^{3,0}_{3}\) |
| \(\{\mathbf{1},\mathbf{2},\mathbf{4},\mathbf{5},\mathbf{6},\mathbf{7}\}\) | \(\text{SU(2})_5:\ \text{FR}^{6,0}_{6}\) |
Quantum Dimensions
| Particle | Numeric | Symbolic |
|---|---|---|
| \(\mathbf{1}\) | \(1.\) | \(1\) |
| \(\mathbf{2}\) | \(1.\) | \(1\) |
| \(\mathbf{3}\) | \(1.41421\) | \(\sqrt{2}\) |
| \(\mathbf{4}\) | \(1.80194\) | \(\text{Root}\left[x^3-x^2-2 x+1,3\right]\) |
| \(\mathbf{5}\) | \(1.80194\) | \(\text{Root}\left[x^3-x^2-2 x+1,3\right]\) |
| \(\mathbf{6}\) | \(2.24698\) | \(\text{Root}\left[x^3-2 x^2-x+1,3\right]\) |
| \(\mathbf{7}\) | \(2.24698\) | \(\text{Root}\left[x^3-2 x^2-x+1,3\right]\) |
| \(\mathbf{8}\) | \(2.54832\) | \(\sqrt{\text{Root}\left[x^3-10 x^2+24 x-8,3\right]}\) |
| \(\mathbf{9}\) | \(3.17771\) | \(\sqrt{\text{Root}\left[x^3-12 x^2+20 x-8,3\right]}\) |
| \(\mathcal{D}_{FP}^2\) | \(37.1836\) | \(\text{Root}\left[x^3-12 x^2+20 x-8,3\right]+2 \text{Root}\left[x^3-x^2-2 x+1,3\right]^2+2 \text{Root}\left[x^3-2 x^2-x+1,3\right]^2+\text{Root}\left[x^3-10 x^2+24 x-8,3\right]+4\) |
Characters
The symbolic character table is the following
\[\begin{array}{|ccccccccc|} \hline \mathbf{1} & \mathbf{2} & \mathbf{3} & \mathbf{5} & \mathbf{4} & \mathbf{6} & \mathbf{7} & \mathbf{8} & \mathbf{9} \\ \hline 1 & 1 & \sqrt{2} & \text{Root}\left[x^3-x^2-2 x+1,3\right] & \text{Root}\left[x^3-x^2-2 x+1,3\right] & \text{Root}\left[x^3-2 x^2-x+1,3\right] & \text{Root}\left[x^3-2 x^2-x+1,3\right] & \text{Root}\left[x^6-10 x^4+24 x^2-8,6\right] & \text{Root}\left[x^6-12 x^4+20 x^2-8,6\right] \\ 1 & 1 & -\sqrt{2} & \text{Root}\left[x^3-x^2-2 x+1,3\right] & \text{Root}\left[x^3-x^2-2 x+1,3\right] & \text{Root}\left[x^3-2 x^2-x+1,3\right] & \text{Root}\left[x^3-2 x^2-x+1,3\right] & \text{Root}\left[x^6-10 x^4+24 x^2-8,1\right] & \text{Root}\left[x^6-12 x^4+20 x^2-8,1\right] \\ 1 & 1 & \sqrt{2} & \text{Root}\left[x^3-x^2-2 x+1,2\right] & \text{Root}\left[x^3-x^2-2 x+1,2\right] & \text{Root}\left[x^3-2 x^2-x+1,1\right] & \text{Root}\left[x^3-2 x^2-x+1,1\right] & \text{Root}\left[x^6-10 x^4+24 x^2-8,4\right] & \text{Root}\left[x^6-12 x^4+20 x^2-8,2\right] \\ 1 & 1 & \sqrt{2} & \text{Root}\left[x^3-x^2-2 x+1,1\right] & \text{Root}\left[x^3-x^2-2 x+1,1\right] & \text{Root}\left[x^3-2 x^2-x+1,2\right] & \text{Root}\left[x^3-2 x^2-x+1,2\right] & \text{Root}\left[x^6-10 x^4+24 x^2-8,2\right] & \text{Root}\left[x^6-12 x^4+20 x^2-8,4\right] \\ 1 & 1 & -\sqrt{2} & \text{Root}\left[x^3-x^2-2 x+1,2\right] & \text{Root}\left[x^3-x^2-2 x+1,2\right] & \text{Root}\left[x^3-2 x^2-x+1,1\right] & \text{Root}\left[x^3-2 x^2-x+1,1\right] & \text{Root}\left[x^6-10 x^4+24 x^2-8,3\right] & \text{Root}\left[x^6-12 x^4+20 x^2-8,5\right] \\ 1 & 1 & -\sqrt{2} & \text{Root}\left[x^3-x^2-2 x+1,1\right] & \text{Root}\left[x^3-x^2-2 x+1,1\right] & \text{Root}\left[x^3-2 x^2-x+1,2\right] & \text{Root}\left[x^3-2 x^2-x+1,2\right] & \text{Root}\left[x^6-10 x^4+24 x^2-8,5\right] & \text{Root}\left[x^6-12 x^4+20 x^2-8,3\right] \\ 1 & -1 & 0 & \text{Root}\left[x^3+x^2-2 x-1,3\right] & \text{Root}\left[x^3-x^2-2 x+1,1\right] & \text{Root}\left[x^3-2 x^2-x+1,2\right] & \text{Root}\left[x^3+2 x^2-x-1,2\right] & 0 & 0 \\ 1 & -1 & 0 & \text{Root}\left[x^3+x^2-2 x-1,1\right] & \text{Root}\left[x^3-x^2-2 x+1,3\right] & \text{Root}\left[x^3-2 x^2-x+1,3\right] & \text{Root}\left[x^3+2 x^2-x-1,1\right] & 0 & 0 \\ 1 & -1 & 0 & \text{Root}\left[x^3+x^2-2 x-1,2\right] & \text{Root}\left[x^3-x^2-2 x+1,2\right] & \text{Root}\left[x^3-2 x^2-x+1,1\right] & \text{Root}\left[x^3+2 x^2-x-1,3\right] & 0 & 0 \\ \hline \end{array}\]The numeric character table is the following
\[\begin{array}{|rrrrrrrrr|} \hline \mathbf{1} & \mathbf{2} & \mathbf{3} & \mathbf{5} & \mathbf{4} & \mathbf{6} & \mathbf{7} & \mathbf{8} & \mathbf{9} \\ \hline 1.000 & 1.000 & 1.414 & 1.802 & 1.802 & 2.247 & 2.247 & 2.548 & 3.178 \\ 1.000 & 1.000 & -1.414 & 1.802 & 1.802 & 2.247 & 2.247 & -2.548 & -3.178 \\ 1.000 & 1.000 & 1.414 & 0.4450 & 0.4450 & -0.8019 & -0.8019 & 0.6294 & -1.134 \\ 1.000 & 1.000 & 1.414 & -1.247 & -1.247 & 0.5550 & 0.5550 & -1.763 & 0.7848 \\ 1.000 & 1.000 & -1.414 & 0.4450 & 0.4450 & -0.8019 & -0.8019 & -0.6294 & 1.134 \\ 1.000 & 1.000 & -1.414 & -1.247 & -1.247 & 0.5550 & 0.5550 & 1.763 & -0.7848 \\ 1.000 & -1.000 & 0 & 1.247 & -1.247 & 0.5550 & -0.5550 & 0 & 0 \\ 1.000 & -1.000 & 0 & -1.802 & 1.802 & 2.247 & -2.247 & 0 & 0 \\ 1.000 & -1.000 & 0 & -0.4450 & 0.4450 & -0.8019 & 0.8019 & 0 & 0 \\ \hline \end{array}\]Modular Data
The matching \(S\)-matrices and twist factors are the following
| \(S\)-matrix | Twist factors |
|---|---|
| \(\frac{1}{\sqrt{\text{Root}\left[x^3-12 x^2+20 x-8,3\right]+2 \text{Root}\left[x^3-x^2-2 x+1,3\right]^2+2 \text{Root}\left[x^3-2 x^2-x+1,3\right]^2+\text{Root}\left[x^3-10 x^2+24 x-8,3\right]+4}}\left(\begin{array}{ccccccccc} \text{Root}\left[56 x^3+28 x^2-1,3\right] \sqrt{\text{Root}\left[x^3-12 x^2+20 x-8,3\right]+2 \text{Root}\left[x^3-x^2-2 x+1,3\right]^2+2 \text{Root}\left[x^3-2 x^2-x+1,3\right]^2+\text{Root}\left[x^3-10 x^2+24 x-8,3\right]+4} & \text{Root}\left[56 x^3+28 x^2-1,3\right] \sqrt{\text{Root}\left[x^3-12 x^2+20 x-8,3\right]+2 \text{Root}\left[x^3-x^2-2 x+1,3\right]^2+2 \text{Root}\left[x^3-2 x^2-x+1,3\right]^2+\text{Root}\left[x^3-10 x^2+24 x-8,3\right]+4} & \text{Root}\left[392 x^6-196 x^4+28 x^2-1,4\right] \sqrt{\text{Root}\left[x^3-12 x^2+20 x-8,3\right]+2 \text{Root}\left[x^3-x^2-2 x+1,3\right]^2+2 \text{Root}\left[x^3-2 x^2-x+1,3\right]^2+\text{Root}\left[x^3-10 x^2+24 x-8,3\right]+4} & \text{Root}\left[56 x^3-28 x^2+1,2\right] \sqrt{\text{Root}\left[x^3-12 x^2+20 x-8,3\right]+2 \text{Root}\left[x^3-x^2-2 x+1,3\right]^2+2 \text{Root}\left[x^3-2 x^2-x+1,3\right]^2+\text{Root}\left[x^3-10 x^2+24 x-8,3\right]+4} & \text{Root}\left[56 x^3-28 x^2+1,2\right] \sqrt{\text{Root}\left[x^3-12 x^2+20 x-8,3\right]+2 \text{Root}\left[x^3-x^2-2 x+1,3\right]^2+2 \text{Root}\left[x^3-2 x^2-x+1,3\right]^2+\text{Root}\left[x^3-10 x^2+24 x-8,3\right]+4} & \text{Root}\left[56 x^3-28 x^2+1,3\right] \sqrt{\text{Root}\left[x^3-12 x^2+20 x-8,3\right]+2 \text{Root}\left[x^3-x^2-2 x+1,3\right]^2+2 \text{Root}\left[x^3-2 x^2-x+1,3\right]^2+\text{Root}\left[x^3-10 x^2+24 x-8,3\right]+4} & \text{Root}\left[56 x^3-28 x^2+1,3\right] \sqrt{\text{Root}\left[x^3-12 x^2+20 x-8,3\right]+2 \text{Root}\left[x^3-x^2-2 x+1,3\right]^2+2 \text{Root}\left[x^3-2 x^2-x+1,3\right]^2+\text{Root}\left[x^3-10 x^2+24 x-8,3\right]+4} & \text{Root}\left[392 x^6-196 x^4+28 x^2-1,5\right] \sqrt{\text{Root}\left[x^3-12 x^2+20 x-8,3\right]+2 \text{Root}\left[x^3-x^2-2 x+1,3\right]^2+2 \text{Root}\left[x^3-2 x^2-x+1,3\right]^2+\text{Root}\left[x^3-10 x^2+24 x-8,3\right]+4} & \text{Root}\left[392 x^6-196 x^4+28 x^2-1,6\right] \sqrt{\text{Root}\left[x^3-12 x^2+20 x-8,3\right]+2 \text{Root}\left[x^3-x^2-2 x+1,3\right]^2+2 \text{Root}\left[x^3-2 x^2-x+1,3\right]^2+\text{Root}\left[x^3-10 x^2+24 x-8,3\right]+4} \\ \text{Root}\left[56 x^3+28 x^2-1,3\right] \sqrt{\text{Root}\left[x^3-12 x^2+20 x-8,3\right]+2 \text{Root}\left[x^3-x^2-2 x+1,3\right]^2+2 \text{Root}\left[x^3-2 x^2-x+1,3\right]^2+\text{Root}\left[x^3-10 x^2+24 x-8,3\right]+4} & \text{Root}\left[56 x^3+28 x^2-1,3\right] \sqrt{\text{Root}\left[x^3-12 x^2+20 x-8,3\right]+2 \text{Root}\left[x^3-x^2-2 x+1,3\right]^2+2 \text{Root}\left[x^3-2 x^2-x+1,3\right]^2+\text{Root}\left[x^3-10 x^2+24 x-8,3\right]+4} & \text{Root}\left[392 x^6-196 x^4+28 x^2-1,3\right] \sqrt{\text{Root}\left[x^3-12 x^2+20 x-8,3\right]+2 \text{Root}\left[x^3-x^2-2 x+1,3\right]^2+2 \text{Root}\left[x^3-2 x^2-x+1,3\right]^2+\text{Root}\left[x^3-10 x^2+24 x-8,3\right]+4} & \text{Root}\left[56 x^3-28 x^2+1,2\right] \sqrt{\text{Root}\left[x^3-12 x^2+20 x-8,3\right]+2 \text{Root}\left[x^3-x^2-2 x+1,3\right]^2+2 \text{Root}\left[x^3-2 x^2-x+1,3\right]^2+\text{Root}\left[x^3-10 x^2+24 x-8,3\right]+4} & \text{Root}\left[56 x^3-28 x^2+1,2\right] \sqrt{\text{Root}\left[x^3-12 x^2+20 x-8,3\right]+2 \text{Root}\left[x^3-x^2-2 x+1,3\right]^2+2 \text{Root}\left[x^3-2 x^2-x+1,3\right]^2+\text{Root}\left[x^3-10 x^2+24 x-8,3\right]+4} & \text{Root}\left[56 x^3-28 x^2+1,3\right] \sqrt{\text{Root}\left[x^3-12 x^2+20 x-8,3\right]+2 \text{Root}\left[x^3-x^2-2 x+1,3\right]^2+2 \text{Root}\left[x^3-2 x^2-x+1,3\right]^2+\text{Root}\left[x^3-10 x^2+24 x-8,3\right]+4} & \text{Root}\left[56 x^3-28 x^2+1,3\right] \sqrt{\text{Root}\left[x^3-12 x^2+20 x-8,3\right]+2 \text{Root}\left[x^3-x^2-2 x+1,3\right]^2+2 \text{Root}\left[x^3-2 x^2-x+1,3\right]^2+\text{Root}\left[x^3-10 x^2+24 x-8,3\right]+4} & \text{Root}\left[392 x^6-196 x^4+28 x^2-1,2\right] \sqrt{\text{Root}\left[x^3-12 x^2+20 x-8,3\right]+2 \text{Root}\left[x^3-x^2-2 x+1,3\right]^2+2 \text{Root}\left[x^3-2 x^2-x+1,3\right]^2+\text{Root}\left[x^3-10 x^2+24 x-8,3\right]+4} & \text{Root}\left[392 x^6-196 x^4+28 x^2-1,1\right] \sqrt{\text{Root}\left[x^3-12 x^2+20 x-8,3\right]+2 \text{Root}\left[x^3-x^2-2 x+1,3\right]^2+2 \text{Root}\left[x^3-2 x^2-x+1,3\right]^2+\text{Root}\left[x^3-10 x^2+24 x-8,3\right]+4} \\ \text{Root}\left[392 x^6-196 x^4+28 x^2-1,4\right] \sqrt{\text{Root}\left[x^3-12 x^2+20 x-8,3\right]+2 \text{Root}\left[x^3-x^2-2 x+1,3\right]^2+2 \text{Root}\left[x^3-2 x^2-x+1,3\right]^2+\text{Root}\left[x^3-10 x^2+24 x-8,3\right]+4} & \text{Root}\left[392 x^6-196 x^4+28 x^2-1,3\right] \sqrt{\text{Root}\left[x^3-12 x^2+20 x-8,3\right]+2 \text{Root}\left[x^3-x^2-2 x+1,3\right]^2+2 \text{Root}\left[x^3-2 x^2-x+1,3\right]^2+\text{Root}\left[x^3-10 x^2+24 x-8,3\right]+4} & 0 & \text{Root}\left[392 x^6-196 x^4+28 x^2-1,5\right] \sqrt{\text{Root}\left[x^3-12 x^2+20 x-8,3\right]+2 \text{Root}\left[x^3-x^2-2 x+1,3\right]^2+2 \text{Root}\left[x^3-2 x^2-x+1,3\right]^2+\text{Root}\left[x^3-10 x^2+24 x-8,3\right]+4} & \text{Root}\left[392 x^6-196 x^4+28 x^2-1,2\right] \sqrt{\text{Root}\left[x^3-12 x^2+20 x-8,3\right]+2 \text{Root}\left[x^3-x^2-2 x+1,3\right]^2+2 \text{Root}\left[x^3-2 x^2-x+1,3\right]^2+\text{Root}\left[x^3-10 x^2+24 x-8,3\right]+4} & \text{Root}\left[392 x^6-196 x^4+28 x^2-1,6\right] \sqrt{\text{Root}\left[x^3-12 x^2+20 x-8,3\right]+2 \text{Root}\left[x^3-x^2-2 x+1,3\right]^2+2 \text{Root}\left[x^3-2 x^2-x+1,3\right]^2+\text{Root}\left[x^3-10 x^2+24 x-8,3\right]+4} & \text{Root}\left[392 x^6-196 x^4+28 x^2-1,1\right] \sqrt{\text{Root}\left[x^3-12 x^2+20 x-8,3\right]+2 \text{Root}\left[x^3-x^2-2 x+1,3\right]^2+2 \text{Root}\left[x^3-2 x^2-x+1,3\right]^2+\text{Root}\left[x^3-10 x^2+24 x-8,3\right]+4} & 0 & 0 \\ \text{Root}\left[56 x^3-28 x^2+1,2\right] \sqrt{\text{Root}\left[x^3-12 x^2+20 x-8,3\right]+2 \text{Root}\left[x^3-x^2-2 x+1,3\right]^2+2 \text{Root}\left[x^3-2 x^2-x+1,3\right]^2+\text{Root}\left[x^3-10 x^2+24 x-8,3\right]+4} & \text{Root}\left[56 x^3-28 x^2+1,2\right] \sqrt{\text{Root}\left[x^3-12 x^2+20 x-8,3\right]+2 \text{Root}\left[x^3-x^2-2 x+1,3\right]^2+2 \text{Root}\left[x^3-2 x^2-x+1,3\right]^2+\text{Root}\left[x^3-10 x^2+24 x-8,3\right]+4} & \text{Root}\left[392 x^6-196 x^4+28 x^2-1,5\right] \sqrt{\text{Root}\left[x^3-12 x^2+20 x-8,3\right]+2 \text{Root}\left[x^3-x^2-2 x+1,3\right]^2+2 \text{Root}\left[x^3-2 x^2-x+1,3\right]^2+\text{Root}\left[x^3-10 x^2+24 x-8,3\right]+4} & \text{Root}\left[56 x^3+28 x^2-1,1\right] \sqrt{\text{Root}\left[x^3-12 x^2+20 x-8,3\right]+2 \text{Root}\left[x^3-x^2-2 x+1,3\right]^2+2 \text{Root}\left[x^3-2 x^2-x+1,3\right]^2+\text{Root}\left[x^3-10 x^2+24 x-8,3\right]+4} & \text{Root}\left[56 x^3+28 x^2-1,1\right] \sqrt{\text{Root}\left[x^3-12 x^2+20 x-8,3\right]+2 \text{Root}\left[x^3-x^2-2 x+1,3\right]^2+2 \text{Root}\left[x^3-2 x^2-x+1,3\right]^2+\text{Root}\left[x^3-10 x^2+24 x-8,3\right]+4} & \text{Root}\left[56 x^3+28 x^2-1,3\right] \sqrt{\text{Root}\left[x^3-12 x^2+20 x-8,3\right]+2 \text{Root}\left[x^3-x^2-2 x+1,3\right]^2+2 \text{Root}\left[x^3-2 x^2-x+1,3\right]^2+\text{Root}\left[x^3-10 x^2+24 x-8,3\right]+4} & \text{Root}\left[56 x^3+28 x^2-1,3\right] \sqrt{\text{Root}\left[x^3-12 x^2+20 x-8,3\right]+2 \text{Root}\left[x^3-x^2-2 x+1,3\right]^2+2 \text{Root}\left[x^3-2 x^2-x+1,3\right]^2+\text{Root}\left[x^3-10 x^2+24 x-8,3\right]+4} & \text{Root}\left[392 x^6-196 x^4+28 x^2-1,1\right] \sqrt{\text{Root}\left[x^3-12 x^2+20 x-8,3\right]+2 \text{Root}\left[x^3-x^2-2 x+1,3\right]^2+2 \text{Root}\left[x^3-2 x^2-x+1,3\right]^2+\text{Root}\left[x^3-10 x^2+24 x-8,3\right]+4} & \text{Root}\left[392 x^6-196 x^4+28 x^2-1,4\right] \sqrt{\text{Root}\left[x^3-12 x^2+20 x-8,3\right]+2 \text{Root}\left[x^3-x^2-2 x+1,3\right]^2+2 \text{Root}\left[x^3-2 x^2-x+1,3\right]^2+\text{Root}\left[x^3-10 x^2+24 x-8,3\right]+4} \\ \text{Root}\left[56 x^3-28 x^2+1,2\right] \sqrt{\text{Root}\left[x^3-12 x^2+20 x-8,3\right]+2 \text{Root}\left[x^3-x^2-2 x+1,3\right]^2+2 \text{Root}\left[x^3-2 x^2-x+1,3\right]^2+\text{Root}\left[x^3-10 x^2+24 x-8,3\right]+4} & \text{Root}\left[56 x^3-28 x^2+1,2\right] \sqrt{\text{Root}\left[x^3-12 x^2+20 x-8,3\right]+2 \text{Root}\left[x^3-x^2-2 x+1,3\right]^2+2 \text{Root}\left[x^3-2 x^2-x+1,3\right]^2+\text{Root}\left[x^3-10 x^2+24 x-8,3\right]+4} & \text{Root}\left[392 x^6-196 x^4+28 x^2-1,2\right] \sqrt{\text{Root}\left[x^3-12 x^2+20 x-8,3\right]+2 \text{Root}\left[x^3-x^2-2 x+1,3\right]^2+2 \text{Root}\left[x^3-2 x^2-x+1,3\right]^2+\text{Root}\left[x^3-10 x^2+24 x-8,3\right]+4} & \text{Root}\left[56 x^3+28 x^2-1,1\right] \sqrt{\text{Root}\left[x^3-12 x^2+20 x-8,3\right]+2 \text{Root}\left[x^3-x^2-2 x+1,3\right]^2+2 \text{Root}\left[x^3-2 x^2-x+1,3\right]^2+\text{Root}\left[x^3-10 x^2+24 x-8,3\right]+4} & \text{Root}\left[56 x^3+28 x^2-1,1\right] \sqrt{\text{Root}\left[x^3-12 x^2+20 x-8,3\right]+2 \text{Root}\left[x^3-x^2-2 x+1,3\right]^2+2 \text{Root}\left[x^3-2 x^2-x+1,3\right]^2+\text{Root}\left[x^3-10 x^2+24 x-8,3\right]+4} & \text{Root}\left[56 x^3+28 x^2-1,3\right] \sqrt{\text{Root}\left[x^3-12 x^2+20 x-8,3\right]+2 \text{Root}\left[x^3-x^2-2 x+1,3\right]^2+2 \text{Root}\left[x^3-2 x^2-x+1,3\right]^2+\text{Root}\left[x^3-10 x^2+24 x-8,3\right]+4} & \text{Root}\left[56 x^3+28 x^2-1,3\right] \sqrt{\text{Root}\left[x^3-12 x^2+20 x-8,3\right]+2 \text{Root}\left[x^3-x^2-2 x+1,3\right]^2+2 \text{Root}\left[x^3-2 x^2-x+1,3\right]^2+\text{Root}\left[x^3-10 x^2+24 x-8,3\right]+4} & \text{Root}\left[392 x^6-196 x^4+28 x^2-1,6\right] \sqrt{\text{Root}\left[x^3-12 x^2+20 x-8,3\right]+2 \text{Root}\left[x^3-x^2-2 x+1,3\right]^2+2 \text{Root}\left[x^3-2 x^2-x+1,3\right]^2+\text{Root}\left[x^3-10 x^2+24 x-8,3\right]+4} & \text{Root}\left[392 x^6-196 x^4+28 x^2-1,3\right] \sqrt{\text{Root}\left[x^3-12 x^2+20 x-8,3\right]+2 \text{Root}\left[x^3-x^2-2 x+1,3\right]^2+2 \text{Root}\left[x^3-2 x^2-x+1,3\right]^2+\text{Root}\left[x^3-10 x^2+24 x-8,3\right]+4} \\ \text{Root}\left[56 x^3-28 x^2+1,3\right] \sqrt{\text{Root}\left[x^3-12 x^2+20 x-8,3\right]+2 \text{Root}\left[x^3-x^2-2 x+1,3\right]^2+2 \text{Root}\left[x^3-2 x^2-x+1,3\right]^2+\text{Root}\left[x^3-10 x^2+24 x-8,3\right]+4} & \text{Root}\left[56 x^3-28 x^2+1,3\right] \sqrt{\text{Root}\left[x^3-12 x^2+20 x-8,3\right]+2 \text{Root}\left[x^3-x^2-2 x+1,3\right]^2+2 \text{Root}\left[x^3-2 x^2-x+1,3\right]^2+\text{Root}\left[x^3-10 x^2+24 x-8,3\right]+4} & \text{Root}\left[392 x^6-196 x^4+28 x^2-1,6\right] \sqrt{\text{Root}\left[x^3-12 x^2+20 x-8,3\right]+2 \text{Root}\left[x^3-x^2-2 x+1,3\right]^2+2 \text{Root}\left[x^3-2 x^2-x+1,3\right]^2+\text{Root}\left[x^3-10 x^2+24 x-8,3\right]+4} & \text{Root}\left[56 x^3+28 x^2-1,3\right] \sqrt{\text{Root}\left[x^3-12 x^2+20 x-8,3\right]+2 \text{Root}\left[x^3-x^2-2 x+1,3\right]^2+2 \text{Root}\left[x^3-2 x^2-x+1,3\right]^2+\text{Root}\left[x^3-10 x^2+24 x-8,3\right]+4} & \text{Root}\left[56 x^3+28 x^2-1,3\right] \sqrt{\text{Root}\left[x^3-12 x^2+20 x-8,3\right]+2 \text{Root}\left[x^3-x^2-2 x+1,3\right]^2+2 \text{Root}\left[x^3-2 x^2-x+1,3\right]^2+\text{Root}\left[x^3-10 x^2+24 x-8,3\right]+4} & \text{Root}\left[56 x^3+28 x^2-1,2\right] \sqrt{\text{Root}\left[x^3-12 x^2+20 x-8,3\right]+2 \text{Root}\left[x^3-x^2-2 x+1,3\right]^2+2 \text{Root}\left[x^3-2 x^2-x+1,3\right]^2+\text{Root}\left[x^3-10 x^2+24 x-8,3\right]+4} & \text{Root}\left[56 x^3+28 x^2-1,2\right] \sqrt{\text{Root}\left[x^3-12 x^2+20 x-8,3\right]+2 \text{Root}\left[x^3-x^2-2 x+1,3\right]^2+2 \text{Root}\left[x^3-2 x^2-x+1,3\right]^2+\text{Root}\left[x^3-10 x^2+24 x-8,3\right]+4} & \text{Root}\left[392 x^6-196 x^4+28 x^2-1,4\right] \sqrt{\text{Root}\left[x^3-12 x^2+20 x-8,3\right]+2 \text{Root}\left[x^3-x^2-2 x+1,3\right]^2+2 \text{Root}\left[x^3-2 x^2-x+1,3\right]^2+\text{Root}\left[x^3-10 x^2+24 x-8,3\right]+4} & \text{Root}\left[392 x^6-196 x^4+28 x^2-1,2\right] \sqrt{\text{Root}\left[x^3-12 x^2+20 x-8,3\right]+2 \text{Root}\left[x^3-x^2-2 x+1,3\right]^2+2 \text{Root}\left[x^3-2 x^2-x+1,3\right]^2+\text{Root}\left[x^3-10 x^2+24 x-8,3\right]+4} \\ \text{Root}\left[56 x^3-28 x^2+1,3\right] \sqrt{\text{Root}\left[x^3-12 x^2+20 x-8,3\right]+2 \text{Root}\left[x^3-x^2-2 x+1,3\right]^2+2 \text{Root}\left[x^3-2 x^2-x+1,3\right]^2+\text{Root}\left[x^3-10 x^2+24 x-8,3\right]+4} & \text{Root}\left[56 x^3-28 x^2+1,3\right] \sqrt{\text{Root}\left[x^3-12 x^2+20 x-8,3\right]+2 \text{Root}\left[x^3-x^2-2 x+1,3\right]^2+2 \text{Root}\left[x^3-2 x^2-x+1,3\right]^2+\text{Root}\left[x^3-10 x^2+24 x-8,3\right]+4} & \text{Root}\left[392 x^6-196 x^4+28 x^2-1,1\right] \sqrt{\text{Root}\left[x^3-12 x^2+20 x-8,3\right]+2 \text{Root}\left[x^3-x^2-2 x+1,3\right]^2+2 \text{Root}\left[x^3-2 x^2-x+1,3\right]^2+\text{Root}\left[x^3-10 x^2+24 x-8,3\right]+4} & \text{Root}\left[56 x^3+28 x^2-1,3\right] \sqrt{\text{Root}\left[x^3-12 x^2+20 x-8,3\right]+2 \text{Root}\left[x^3-x^2-2 x+1,3\right]^2+2 \text{Root}\left[x^3-2 x^2-x+1,3\right]^2+\text{Root}\left[x^3-10 x^2+24 x-8,3\right]+4} & \text{Root}\left[56 x^3+28 x^2-1,3\right] \sqrt{\text{Root}\left[x^3-12 x^2+20 x-8,3\right]+2 \text{Root}\left[x^3-x^2-2 x+1,3\right]^2+2 \text{Root}\left[x^3-2 x^2-x+1,3\right]^2+\text{Root}\left[x^3-10 x^2+24 x-8,3\right]+4} & \text{Root}\left[56 x^3+28 x^2-1,2\right] \sqrt{\text{Root}\left[x^3-12 x^2+20 x-8,3\right]+2 \text{Root}\left[x^3-x^2-2 x+1,3\right]^2+2 \text{Root}\left[x^3-2 x^2-x+1,3\right]^2+\text{Root}\left[x^3-10 x^2+24 x-8,3\right]+4} & \text{Root}\left[56 x^3+28 x^2-1,2\right] \sqrt{\text{Root}\left[x^3-12 x^2+20 x-8,3\right]+2 \text{Root}\left[x^3-x^2-2 x+1,3\right]^2+2 \text{Root}\left[x^3-2 x^2-x+1,3\right]^2+\text{Root}\left[x^3-10 x^2+24 x-8,3\right]+4} & \text{Root}\left[392 x^6-196 x^4+28 x^2-1,3\right] \sqrt{\text{Root}\left[x^3-12 x^2+20 x-8,3\right]+2 \text{Root}\left[x^3-x^2-2 x+1,3\right]^2+2 \text{Root}\left[x^3-2 x^2-x+1,3\right]^2+\text{Root}\left[x^3-10 x^2+24 x-8,3\right]+4} & \text{Root}\left[392 x^6-196 x^4+28 x^2-1,5\right] \sqrt{\text{Root}\left[x^3-12 x^2+20 x-8,3\right]+2 \text{Root}\left[x^3-x^2-2 x+1,3\right]^2+2 \text{Root}\left[x^3-2 x^2-x+1,3\right]^2+\text{Root}\left[x^3-10 x^2+24 x-8,3\right]+4} \\ \text{Root}\left[392 x^6-196 x^4+28 x^2-1,5\right] \sqrt{\text{Root}\left[x^3-12 x^2+20 x-8,3\right]+2 \text{Root}\left[x^3-x^2-2 x+1,3\right]^2+2 \text{Root}\left[x^3-2 x^2-x+1,3\right]^2+\text{Root}\left[x^3-10 x^2+24 x-8,3\right]+4} & \text{Root}\left[392 x^6-196 x^4+28 x^2-1,2\right] \sqrt{\text{Root}\left[x^3-12 x^2+20 x-8,3\right]+2 \text{Root}\left[x^3-x^2-2 x+1,3\right]^2+2 \text{Root}\left[x^3-2 x^2-x+1,3\right]^2+\text{Root}\left[x^3-10 x^2+24 x-8,3\right]+4} & 0 & \text{Root}\left[392 x^6-196 x^4+28 x^2-1,1\right] \sqrt{\text{Root}\left[x^3-12 x^2+20 x-8,3\right]+2 \text{Root}\left[x^3-x^2-2 x+1,3\right]^2+2 \text{Root}\left[x^3-2 x^2-x+1,3\right]^2+\text{Root}\left[x^3-10 x^2+24 x-8,3\right]+4} & \text{Root}\left[392 x^6-196 x^4+28 x^2-1,6\right] \sqrt{\text{Root}\left[x^3-12 x^2+20 x-8,3\right]+2 \text{Root}\left[x^3-x^2-2 x+1,3\right]^2+2 \text{Root}\left[x^3-2 x^2-x+1,3\right]^2+\text{Root}\left[x^3-10 x^2+24 x-8,3\right]+4} & \text{Root}\left[392 x^6-196 x^4+28 x^2-1,4\right] \sqrt{\text{Root}\left[x^3-12 x^2+20 x-8,3\right]+2 \text{Root}\left[x^3-x^2-2 x+1,3\right]^2+2 \text{Root}\left[x^3-2 x^2-x+1,3\right]^2+\text{Root}\left[x^3-10 x^2+24 x-8,3\right]+4} & \text{Root}\left[392 x^6-196 x^4+28 x^2-1,3\right] \sqrt{\text{Root}\left[x^3-12 x^2+20 x-8,3\right]+2 \text{Root}\left[x^3-x^2-2 x+1,3\right]^2+2 \text{Root}\left[x^3-2 x^2-x+1,3\right]^2+\text{Root}\left[x^3-10 x^2+24 x-8,3\right]+4} & 0 & 0 \\ \text{Root}\left[392 x^6-196 x^4+28 x^2-1,6\right] \sqrt{\text{Root}\left[x^3-12 x^2+20 x-8,3\right]+2 \text{Root}\left[x^3-x^2-2 x+1,3\right]^2+2 \text{Root}\left[x^3-2 x^2-x+1,3\right]^2+\text{Root}\left[x^3-10 x^2+24 x-8,3\right]+4} & \text{Root}\left[392 x^6-196 x^4+28 x^2-1,1\right] \sqrt{\text{Root}\left[x^3-12 x^2+20 x-8,3\right]+2 \text{Root}\left[x^3-x^2-2 x+1,3\right]^2+2 \text{Root}\left[x^3-2 x^2-x+1,3\right]^2+\text{Root}\left[x^3-10 x^2+24 x-8,3\right]+4} & 0 & \text{Root}\left[392 x^6-196 x^4+28 x^2-1,4\right] \sqrt{\text{Root}\left[x^3-12 x^2+20 x-8,3\right]+2 \text{Root}\left[x^3-x^2-2 x+1,3\right]^2+2 \text{Root}\left[x^3-2 x^2-x+1,3\right]^2+\text{Root}\left[x^3-10 x^2+24 x-8,3\right]+4} & \text{Root}\left[392 x^6-196 x^4+28 x^2-1,3\right] \sqrt{\text{Root}\left[x^3-12 x^2+20 x-8,3\right]+2 \text{Root}\left[x^3-x^2-2 x+1,3\right]^2+2 \text{Root}\left[x^3-2 x^2-x+1,3\right]^2+\text{Root}\left[x^3-10 x^2+24 x-8,3\right]+4} & \text{Root}\left[392 x^6-196 x^4+28 x^2-1,2\right] \sqrt{\text{Root}\left[x^3-12 x^2+20 x-8,3\right]+2 \text{Root}\left[x^3-x^2-2 x+1,3\right]^2+2 \text{Root}\left[x^3-2 x^2-x+1,3\right]^2+\text{Root}\left[x^3-10 x^2+24 x-8,3\right]+4} & \text{Root}\left[392 x^6-196 x^4+28 x^2-1,5\right] \sqrt{\text{Root}\left[x^3-12 x^2+20 x-8,3\right]+2 \text{Root}\left[x^3-x^2-2 x+1,3\right]^2+2 \text{Root}\left[x^3-2 x^2-x+1,3\right]^2+\text{Root}\left[x^3-10 x^2+24 x-8,3\right]+4} & 0 & 0 \\\end{array}\right)\) | \(\begin{array}{l}\left(0,\frac{1}{2},-\frac{7}{16},\frac{1}{7},-\frac{5}{14},-\frac{2}{7},\frac{3}{14},-\frac{33}{112},\frac{31}{112}\right) \\\left(0,\frac{1}{2},\frac{1}{16},-\frac{1}{7},\frac{5}{14},\frac{2}{7},-\frac{3}{14},-\frac{9}{112},\frac{39}{112}\right) \\\left(0,\frac{1}{2},\frac{3}{16},\frac{1}{7},-\frac{5}{14},-\frac{2}{7},\frac{3}{14},\frac{37}{112},-\frac{11}{112}\right) \\\left(0,\frac{1}{2},-\frac{5}{16},-\frac{1}{7},\frac{5}{14},\frac{2}{7},-\frac{3}{14},-\frac{51}{112},-\frac{3}{112}\right) \\\left(0,\frac{1}{2},-\frac{3}{16},\frac{1}{7},-\frac{5}{14},-\frac{2}{7},\frac{3}{14},-\frac{5}{112},-\frac{53}{112}\right) \\\left(0,\frac{1}{2},\frac{5}{16},-\frac{1}{7},\frac{5}{14},\frac{2}{7},-\frac{3}{14},\frac{19}{112},-\frac{45}{112}\right) \\\left(0,\frac{1}{2},\frac{7}{16},\frac{1}{7},-\frac{5}{14},-\frac{2}{7},\frac{3}{14},-\frac{47}{112},\frac{17}{112}\right) \\\left(0,\frac{1}{2},-\frac{1}{16},-\frac{1}{7},\frac{5}{14},\frac{2}{7},-\frac{3}{14},-\frac{23}{112},\frac{25}{112}\right) \\\left(0,\frac{1}{2},\frac{1}{16},\frac{1}{7},-\frac{5}{14},-\frac{2}{7},\frac{3}{14},\frac{23}{112},-\frac{25}{112}\right) \\\left(0,\frac{1}{2},-\frac{7}{16},-\frac{1}{7},\frac{5}{14},\frac{2}{7},-\frac{3}{14},\frac{47}{112},-\frac{17}{112}\right) \\\left(0,\frac{1}{2},-\frac{5}{16},\frac{1}{7},-\frac{5}{14},-\frac{2}{7},\frac{3}{14},-\frac{19}{112},\frac{45}{112}\right) \\\left(0,\frac{1}{2},\frac{3}{16},-\frac{1}{7},\frac{5}{14},\frac{2}{7},-\frac{3}{14},\frac{5}{112},\frac{53}{112}\right) \\\left(0,\frac{1}{2},\frac{5}{16},\frac{1}{7},-\frac{5}{14},-\frac{2}{7},\frac{3}{14},\frac{51}{112},\frac{3}{112}\right) \\\left(0,\frac{1}{2},-\frac{3}{16},-\frac{1}{7},\frac{5}{14},\frac{2}{7},-\frac{3}{14},-\frac{37}{112},\frac{11}{112}\right) \\\left(0,\frac{1}{2},-\frac{1}{16},\frac{1}{7},-\frac{5}{14},-\frac{2}{7},\frac{3}{14},\frac{9}{112},-\frac{39}{112}\right) \\\left(0,\frac{1}{2},\frac{7}{16},-\frac{1}{7},\frac{5}{14},\frac{2}{7},-\frac{3}{14},\frac{33}{112},-\frac{31}{112}\right)\end{array}\) |
Adjoint Subring
Particles \(\mathbf{1}, \mathbf{2}, \mathbf{4}, \mathbf{5}, \mathbf{6}, \mathbf{7}\), form the adjoint subring \(\text{SU(2})_5:\ \text{FR}^{6,0}_{6}\) .
The upper central series is the following: \(\text{Ising$\times $}\text{PSU}(2)_5 \underset{ \mathbf{1}, \mathbf{2}, \mathbf{4}, \mathbf{5}, \mathbf{6}, \mathbf{7} }{\supset} \text{SU(2})_5 \underset{ \mathbf{1}, \mathbf{3}, \mathbf{4} }{\supset} \text{PSU}(2)_5\)
Universal grading
Each particle can be graded as follows: \(\text{deg}(\mathbf{1}) = \mathbf{1}', \text{deg}(\mathbf{2}) = \mathbf{1}', \text{deg}(\mathbf{3}) = \mathbf{2}', \text{deg}(\mathbf{4}) = \mathbf{1}', \text{deg}(\mathbf{5}) = \mathbf{1}', \text{deg}(\mathbf{6}) = \mathbf{1}', \text{deg}(\mathbf{7}) = \mathbf{1}', \text{deg}(\mathbf{8}) = \mathbf{2}', \text{deg}(\mathbf{9}) = \mathbf{2}'\), 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: