\(\text{Rep(}D_3\text{)$\times $}\text{PSU}(2)_5:\ \text{FR}^{9,0}_{20}\)
Fusion Rules
\[\begin{array}{|lllllllll|} \hline \mathbf{1} & \mathbf{2} & \mathbf{3} & \mathbf{4} & \mathbf{5} & \mathbf{6} & \mathbf{7} & \mathbf{8} & \mathbf{9} \\ \mathbf{2} & \mathbf{1} & \mathbf{4} & \mathbf{3} & \mathbf{5} & \mathbf{7} & \mathbf{6} & \mathbf{8} & \mathbf{9} \\ \mathbf{3} & \mathbf{4} & \mathbf{1}+\mathbf{6} & \mathbf{2}+\mathbf{7} & \mathbf{8} & \mathbf{3}+\mathbf{6} & \mathbf{4}+\mathbf{7} & \mathbf{5}+\mathbf{9} & \mathbf{8}+\mathbf{9} \\ \mathbf{4} & \mathbf{3} & \mathbf{2}+\mathbf{7} & \mathbf{1}+\mathbf{6} & \mathbf{8} & \mathbf{4}+\mathbf{7} & \mathbf{3}+\mathbf{6} & \mathbf{5}+\mathbf{9} & \mathbf{8}+\mathbf{9} \\ \mathbf{5} & \mathbf{5} & \mathbf{8} & \mathbf{8} & \mathbf{1}+\mathbf{2}+\mathbf{5} & \mathbf{9} & \mathbf{9} & \mathbf{3}+\mathbf{4}+\mathbf{8} & \mathbf{6}+\mathbf{7}+\mathbf{9} \\ \mathbf{6} & \mathbf{7} & \mathbf{3}+\mathbf{6} & \mathbf{4}+\mathbf{7} & \mathbf{9} & \mathbf{1}+\mathbf{3}+\mathbf{6} & \mathbf{2}+\mathbf{4}+\mathbf{7} & \mathbf{8}+\mathbf{9} & \mathbf{5}+\mathbf{8}+\mathbf{9} \\ \mathbf{7} & \mathbf{6} & \mathbf{4}+\mathbf{7} & \mathbf{3}+\mathbf{6} & \mathbf{9} & \mathbf{2}+\mathbf{4}+\mathbf{7} & \mathbf{1}+\mathbf{3}+\mathbf{6} & \mathbf{8}+\mathbf{9} & \mathbf{5}+\mathbf{8}+\mathbf{9} \\ \mathbf{8} & \mathbf{8} & \mathbf{5}+\mathbf{9} & \mathbf{5}+\mathbf{9} & \mathbf{3}+\mathbf{4}+\mathbf{8} & \mathbf{8}+\mathbf{9} & \mathbf{8}+\mathbf{9} & \mathbf{1}+\mathbf{2}+\mathbf{5}+\mathbf{6}+\mathbf{7}+\mathbf{9} & \mathbf{3}+\mathbf{4}+\mathbf{6}+\mathbf{7}+\mathbf{8}+\mathbf{9} \\ \mathbf{9} & \mathbf{9} & \mathbf{8}+\mathbf{9} & \mathbf{8}+\mathbf{9} & \mathbf{6}+\mathbf{7}+\mathbf{9} & \mathbf{5}+\mathbf{8}+\mathbf{9} & \mathbf{5}+\mathbf{8}+\mathbf{9} & \mathbf{3}+\mathbf{4}+\mathbf{6}+\mathbf{7}+\mathbf{8}+\mathbf{9} & \mathbf{1}+\mathbf{2}+\mathbf{3}+\mathbf{4}+\mathbf{5}+\mathbf{6}+\mathbf{7}+\mathbf{8}+\mathbf{9} \\ \hline \end{array}\]The following particles form non-trivial sub fusion rings
| Particles | SubRing |
|---|---|
| \(\{\mathbf{1},\mathbf{2}\}\) | \(\mathbb{Z}_2:\ \text{FR}^{2,0}_{1}\) |
| \(\{\mathbf{1},\mathbf{2},\mathbf{5}\}\) | \(\left.\text{Rep(}D_3\right):\ \text{FR}^{3,0}_{2}\) |
| \(\{\mathbf{1},\mathbf{3},\mathbf{6}\}\) | \(\text{PSU}(2)_5:\ \text{FR}^{3,0}_{3}\) |
| \(\{\mathbf{1},\mathbf{2},\mathbf{3},\mathbf{4},\mathbf{6},\mathbf{7}\}\) | \(\text{SU(2})_5:\ \text{FR}^{6,0}_{6}\) |
Quantum Dimensions
| Particle | Numeric | Symbolic |
|---|---|---|
| \(\mathbf{1}\) | \(1.\) | \(1\) |
| \(\mathbf{2}\) | \(1.\) | \(1\) |
| \(\mathbf{3}\) | \(1.80194\) | \(\text{Root}\left[x^3-x^2-2 x+1,3\right]\) |
| \(\mathbf{4}\) | \(1.80194\) | \(\text{Root}\left[x^3-x^2-2 x+1,3\right]\) |
| \(\mathbf{5}\) | \(2.\) | \(2\) |
| \(\mathbf{6}\) | \(2.24698\) | \(\text{Root}\left[x^3-2 x^2-x+1,3\right]\) |
| \(\mathbf{7}\) | \(2.24698\) | \(\text{Root}\left[x^3-2 x^2-x+1,3\right]\) |
| \(\mathbf{8}\) | \(3.60388\) | \(\text{Root}\left[x^3-2 x^2-8 x+8,3\right]\) |
| \(\mathbf{9}\) | \(4.49396\) | \(\text{Root}\left[x^3-4 x^2-4 x+8,3\right]\) |
| \(\mathcal{D}_{FP}^2\) | \(55.7754\) | \(2 \text{Root}\left[x^3-x^2-2 x+1,3\right]^2+2 \text{Root}\left[x^3-2 x^2-x+1,3\right]^2+\text{Root}\left[x^3-2 x^2-8 x+8,3\right]^2+\text{Root}\left[x^3-4 x^2-4 x+8,3\right]^2+6\) |
Characters
The symbolic character table is the following
\[\begin{array}{|ccccccccc|} \hline \mathbf{1} & \mathbf{2} & \mathbf{4} & \mathbf{3} & \mathbf{5} & \mathbf{7} & \mathbf{6} & \mathbf{8} & \mathbf{9} \\ \hline 1 & 1 & \text{Root}\left[x^3-x^2-2 x+1,3\right] & \text{Root}\left[x^3-x^2-2 x+1,3\right] & 2 & \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^3-2 x^2-8 x+8,3\right] & \text{Root}\left[x^3-4 x^2-4 x+8,3\right] \\ 1 & 1 & \text{Root}\left[x^3-x^2-2 x+1,3\right] & \text{Root}\left[x^3-x^2-2 x+1,3\right] & -1 & \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^3+x^2-2 x-1,1\right] & \text{Root}\left[x^3+2 x^2-x-1,1\right] \\ 1 & 1 & \text{Root}\left[x^3-x^2-2 x+1,2\right] & \text{Root}\left[x^3-x^2-2 x+1,2\right] & 2 & \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^3-2 x^2-8 x+8,2\right] & \text{Root}\left[x^3-4 x^2-4 x+8,1\right] \\ 1 & 1 & \text{Root}\left[x^3-x^2-2 x+1,1\right] & \text{Root}\left[x^3-x^2-2 x+1,1\right] & 2 & \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^3-2 x^2-8 x+8,1\right] & \text{Root}\left[x^3-4 x^2-4 x+8,2\right] \\ 1 & 1 & \text{Root}\left[x^3-x^2-2 x+1,2\right] & \text{Root}\left[x^3-x^2-2 x+1,2\right] & -1 & \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^3+x^2-2 x-1,2\right] & \text{Root}\left[x^3+2 x^2-x-1,3\right] \\ 1 & 1 & \text{Root}\left[x^3-x^2-2 x+1,1\right] & \text{Root}\left[x^3-x^2-2 x+1,1\right] & -1 & \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^3+x^2-2 x-1,3\right] & \text{Root}\left[x^3+2 x^2-x-1,2\right] \\ 1 & -1 & \text{Root}\left[x^3+x^2-2 x-1,2\right] & \text{Root}\left[x^3-x^2-2 x+1,2\right] & 0 & \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 & \text{Root}\left[x^3+x^2-2 x-1,3\right] & \text{Root}\left[x^3-x^2-2 x+1,1\right] & 0 & \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 & \text{Root}\left[x^3+x^2-2 x-1,1\right] & \text{Root}\left[x^3-x^2-2 x+1,3\right] & 0 & \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
\[\begin{array}{|rrrrrrrrr|} \hline \mathbf{1} & \mathbf{2} & \mathbf{4} & \mathbf{3} & \mathbf{5} & \mathbf{7} & \mathbf{6} & \mathbf{8} & \mathbf{9} \\ \hline 1.000 & 1.000 & 1.802 & 1.802 & 2.000 & 2.247 & 2.247 & 3.604 & 4.494 \\ 1.000 & 1.000 & 1.802 & 1.802 & -1.000 & 2.247 & 2.247 & -1.802 & -2.247 \\ 1.000 & 1.000 & 0.4450 & 0.4450 & 2.000 & -0.8019 & -0.8019 & 0.8901 & -1.604 \\ 1.000 & 1.000 & -1.247 & -1.247 & 2.000 & 0.5550 & 0.5550 & -2.494 & 1.110 \\ 1.000 & 1.000 & 0.4450 & 0.4450 & -1.000 & -0.8019 & -0.8019 & -0.4450 & 0.8019 \\ 1.000 & 1.000 & -1.247 & -1.247 & -1.000 & 0.5550 & 0.5550 & 1.247 & -0.5550 \\ 1.000 & -1.000 & -0.4450 & 0.4450 & 0 & 0.8019 & -0.8019 & 0 & 0 \\ 1.000 & -1.000 & 1.247 & -1.247 & 0 & -0.5550 & 0.5550 & 0 & 0 \\ 1.000 & -1.000 & -1.802 & 1.802 & 0 & -2.247 & 2.247 & 0 & 0 \\ \hline \end{array}\]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
Data
Download links for numeric data: