\(\text{PSU}(2)_{16}:\ \text{FR}^{9,0}_{41}\)
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{1}+\mathbf{4}+\mathbf{6} & \mathbf{2}+\mathbf{3}+\mathbf{5} & \mathbf{4}+\mathbf{6}+\mathbf{8} & \mathbf{3}+\mathbf{5}+\mathbf{7} & \mathbf{6}+\mathbf{8}+\mathbf{9} & \mathbf{5}+\mathbf{7}+\mathbf{9} & \mathbf{7}+\mathbf{8}+\mathbf{9} \\ \mathbf{4} & \mathbf{3} & \mathbf{2}+\mathbf{3}+\mathbf{5} & \mathbf{1}+\mathbf{4}+\mathbf{6} & \mathbf{3}+\mathbf{5}+\mathbf{7} & \mathbf{4}+\mathbf{6}+\mathbf{8} & \mathbf{5}+\mathbf{7}+\mathbf{9} & \mathbf{6}+\mathbf{8}+\mathbf{9} & \mathbf{7}+\mathbf{8}+\mathbf{9} \\ \mathbf{5} & \mathbf{6} & \mathbf{4}+\mathbf{6}+\mathbf{8} & \mathbf{3}+\mathbf{5}+\mathbf{7} & \mathbf{1}+\mathbf{4}+\mathbf{6}+\mathbf{8}+\mathbf{9} & \mathbf{2}+\mathbf{3}+\mathbf{5}+\mathbf{7}+\mathbf{9} & \mathbf{4}+\mathbf{6}+\mathbf{7}+\mathbf{8}+\mathbf{9} & \mathbf{3}+\mathbf{5}+\mathbf{7}+\mathbf{8}+\mathbf{9} & \mathbf{5}+\mathbf{6}+\mathbf{7}+\mathbf{8}+\mathbf{9} \\ \mathbf{6} & \mathbf{5} & \mathbf{3}+\mathbf{5}+\mathbf{7} & \mathbf{4}+\mathbf{6}+\mathbf{8} & \mathbf{2}+\mathbf{3}+\mathbf{5}+\mathbf{7}+\mathbf{9} & \mathbf{1}+\mathbf{4}+\mathbf{6}+\mathbf{8}+\mathbf{9} & \mathbf{3}+\mathbf{5}+\mathbf{7}+\mathbf{8}+\mathbf{9} & \mathbf{4}+\mathbf{6}+\mathbf{7}+\mathbf{8}+\mathbf{9} & \mathbf{5}+\mathbf{6}+\mathbf{7}+\mathbf{8}+\mathbf{9} \\ \mathbf{7} & \mathbf{8} & \mathbf{6}+\mathbf{8}+\mathbf{9} & \mathbf{5}+\mathbf{7}+\mathbf{9} & \mathbf{4}+\mathbf{6}+\mathbf{7}+\mathbf{8}+\mathbf{9} & \mathbf{3}+\mathbf{5}+\mathbf{7}+\mathbf{8}+\mathbf{9} & \mathbf{1}+\mathbf{4}+\mathbf{5}+\mathbf{6}+\mathbf{7}+\mathbf{8}+\mathbf{9} & \mathbf{2}+\mathbf{3}+\mathbf{5}+\mathbf{6}+\mathbf{7}+\mathbf{8}+\mathbf{9} & \mathbf{3}+\mathbf{4}+\mathbf{5}+\mathbf{6}+\mathbf{7}+\mathbf{8}+\mathbf{9} \\ \mathbf{8} & \mathbf{7} & \mathbf{5}+\mathbf{7}+\mathbf{9} & \mathbf{6}+\mathbf{8}+\mathbf{9} & \mathbf{3}+\mathbf{5}+\mathbf{7}+\mathbf{8}+\mathbf{9} & \mathbf{4}+\mathbf{6}+\mathbf{7}+\mathbf{8}+\mathbf{9} & \mathbf{2}+\mathbf{3}+\mathbf{5}+\mathbf{6}+\mathbf{7}+\mathbf{8}+\mathbf{9} & \mathbf{1}+\mathbf{4}+\mathbf{5}+\mathbf{6}+\mathbf{7}+\mathbf{8}+\mathbf{9} & \mathbf{3}+\mathbf{4}+\mathbf{5}+\mathbf{6}+\mathbf{7}+\mathbf{8}+\mathbf{9} \\ \mathbf{9} & \mathbf{9} & \mathbf{7}+\mathbf{8}+\mathbf{9} & \mathbf{7}+\mathbf{8}+\mathbf{9} & \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{5}+\mathbf{6}+\mathbf{7}+\mathbf{8}+\mathbf{9} & \mathbf{3}+\mathbf{4}+\mathbf{5}+\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 |
|---|---|
| \(\{\mathbf{1},\mathbf{2}\}\) | \(\mathbb{Z}_2:\ \text{FR}^{2,0}_{1}\) |
Quantum Dimensions
| Particle | Numeric | Symbolic |
|---|---|---|
| \(\mathbf{1}\) | \(1.\) | \(1\) |
| \(\mathbf{2}\) | \(1.\) | \(1\) |
| \(\mathbf{3}\) | \(2.87939\) | \(\text{Root}\left[x^3-3 x^2+1,3\right]\) |
| \(\mathbf{4}\) | \(2.87939\) | \(\text{Root}\left[x^3-3 x^2+1,3\right]\) |
| \(\mathbf{5}\) | \(4.41147\) | \(\text{Root}\left[x^3-3 x^2-6 x-1,3\right]\) |
| \(\mathbf{6}\) | \(4.41147\) | \(\text{Root}\left[x^3-3 x^2-6 x-1,3\right]\) |
| \(\mathbf{7}\) | \(5.41147\) | \(\text{Root}\left[x^3-6 x^2+3 x+1,3\right]\) |
| \(\mathbf{8}\) | \(5.41147\) | \(\text{Root}\left[x^3-6 x^2+3 x+1,3\right]\) |
| \(\mathbf{9}\) | \(5.75877\) | \(\text{Root}\left[x^3-6 x^2+8,3\right]\) |
| \(\mathcal{D}_{FP}^2\) | \(149.235\) | \(2 \text{Root}\left[x^3-3 x^2+1,3\right]^2+2 \text{Root}\left[x^3-3 x^2-6 x-1,3\right]^2+2 \text{Root}\left[x^3-6 x^2+3 x+1,3\right]^2+\text{Root}\left[x^3-6 x^2+8,3\right]^2+2\) |
Characters
The symbolic character table is the following
\[\begin{array}{|ccccccccc|} \hline \mathbf{1} & \mathbf{2} & \mathbf{4} & \mathbf{3} & \mathbf{5} & \mathbf{6} & \mathbf{8} & \mathbf{7} & \mathbf{9} \\ \hline 1 & 1 & \text{Root}\left[x^3-3 x^2+1,3\right] & \text{Root}\left[x^3-3 x^2+1,3\right] & \text{Root}\left[x^3-3 x^2-6 x-1,3\right] & \text{Root}\left[x^3-3 x^2-6 x-1,3\right] & \text{Root}\left[x^3-6 x^2+3 x+1,3\right] & \text{Root}\left[x^3-6 x^2+3 x+1,3\right] & \text{Root}\left[x^3-6 x^2+8,3\right] \\ 1 & 1 & 2 & 2 & 1 & 1 & -1 & -1 & -2 \\ 1 & 1 & \text{Root}\left[x^3-3 x^2+1,2\right] & \text{Root}\left[x^3-3 x^2+1,2\right] & \text{Root}\left[x^3-3 x^2-6 x-1,1\right] & \text{Root}\left[x^3-3 x^2-6 x-1,1\right] & \text{Root}\left[x^3-6 x^2+3 x+1,1\right] & \text{Root}\left[x^3-6 x^2+3 x+1,1\right] & \text{Root}\left[x^3-6 x^2+8,2\right] \\ 1 & 1 & \text{Root}\left[x^3-3 x^2+1,1\right] & \text{Root}\left[x^3-3 x^2+1,1\right] & \text{Root}\left[x^3-3 x^2-6 x-1,2\right] & \text{Root}\left[x^3-3 x^2-6 x-1,2\right] & \text{Root}\left[x^3-6 x^2+3 x+1,2\right] & \text{Root}\left[x^3-6 x^2+3 x+1,2\right] & \text{Root}\left[x^3-6 x^2+8,1\right] \\ 1 & 1 & -1 & -1 & 1 & 1 & -1 & -1 & 1 \\ 1 & -1 & \text{Root}\left[x^3-3 x^2+3,3\right] & \text{Root}\left[x^3+3 x^2-3,1\right] & \text{Root}\left[x^3+3 x^2-1,1\right] & \text{Root}\left[x^3-3 x^2+1,3\right] & \text{Root}\left[x^3-3 x-1,3\right] & \text{Root}\left[x^3-3 x+1,1\right] & 0 \\ 1 & -1 & 0 & 0 & 1 & -1 & 1 & -1 & 0 \\ 1 & -1 & \text{Root}\left[x^3-3 x^2+3,2\right] & \text{Root}\left[x^3+3 x^2-3,2\right] & \text{Root}\left[x^3+3 x^2-1,3\right] & \text{Root}\left[x^3-3 x^2+1,1\right] & \text{Root}\left[x^3-3 x-1,1\right] & \text{Root}\left[x^3-3 x+1,3\right] & 0 \\ 1 & -1 & \text{Root}\left[x^3-3 x^2+3,1\right] & \text{Root}\left[x^3+3 x^2-3,3\right] & \text{Root}\left[x^3+3 x^2-1,2\right] & \text{Root}\left[x^3-3 x^2+1,2\right] & \text{Root}\left[x^3-3 x-1,2\right] & \text{Root}\left[x^3-3 x+1,2\right] & 0 \\ \hline \end{array}\]The numeric character table is the following
\[\begin{array}{|rrrrrrrrr|} \hline \mathbf{1} & \mathbf{2} & \mathbf{4} & \mathbf{3} & \mathbf{5} & \mathbf{6} & \mathbf{8} & \mathbf{7} & \mathbf{9} \\ \hline 1.000 & 1.000 & 2.879 & 2.879 & 4.411 & 4.411 & 5.411 & 5.411 & 5.759 \\ 1.000 & 1.000 & 2.000 & 2.000 & 1.000 & 1.000 & -1.000 & -1.000 & -2.000 \\ 1.000 & 1.000 & 0.6527 & 0.6527 & -1.227 & -1.227 & -0.2267 & -0.2267 & 1.305 \\ 1.000 & 1.000 & -0.5321 & -0.5321 & -0.1848 & -0.1848 & 0.8152 & 0.8152 & -1.064 \\ 1.000 & 1.000 & -1.000 & -1.000 & 1.000 & 1.000 & -1.000 & -1.000 & 1.000 \\ 1.000 & -1.000 & 2.532 & -2.532 & -2.879 & 2.879 & 1.879 & -1.879 & 0 \\ 1.000 & -1.000 & 0 & 0 & 1.000 & -1.000 & 1.000 & -1.000 & 0 \\ 1.000 & -1.000 & 1.347 & -1.347 & 0.5321 & -0.5321 & -1.532 & 1.532 & 0 \\ 1.000 & -1.000 & -0.8794 & 0.8794 & -0.6527 & 0.6527 & -0.3473 & 0.3473 & 0 \\ \hline \end{array}\]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
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