\(\text{PSU}(2)_{12}:\ \text{FR}^{7,0}_{14}\)
Fusion Rules
\[\begin{array}{|lllllll|} \hline \mathbf{1} & \mathbf{2} & \mathbf{3} & \mathbf{4} & \mathbf{5} & \mathbf{6} & \mathbf{7} \\ \mathbf{2} & \mathbf{1} & \mathbf{4} & \mathbf{3} & \mathbf{6} & \mathbf{5} & \mathbf{7} \\ \mathbf{3} & \mathbf{4} & \mathbf{1}+\mathbf{4}+\mathbf{6} & \mathbf{2}+\mathbf{3}+\mathbf{5} & \mathbf{4}+\mathbf{6}+\mathbf{7} & \mathbf{3}+\mathbf{5}+\mathbf{7} & \mathbf{5}+\mathbf{6}+\mathbf{7} \\ \mathbf{4} & \mathbf{3} & \mathbf{2}+\mathbf{3}+\mathbf{5} & \mathbf{1}+\mathbf{4}+\mathbf{6} & \mathbf{3}+\mathbf{5}+\mathbf{7} & \mathbf{4}+\mathbf{6}+\mathbf{7} & \mathbf{5}+\mathbf{6}+\mathbf{7} \\ \mathbf{5} & \mathbf{6} & \mathbf{4}+\mathbf{6}+\mathbf{7} & \mathbf{3}+\mathbf{5}+\mathbf{7} & \mathbf{1}+\mathbf{4}+\mathbf{5}+\mathbf{6}+\mathbf{7} & \mathbf{2}+\mathbf{3}+\mathbf{5}+\mathbf{6}+\mathbf{7} & \mathbf{3}+\mathbf{4}+\mathbf{5}+\mathbf{6}+\mathbf{7} \\ \mathbf{6} & \mathbf{5} & \mathbf{3}+\mathbf{5}+\mathbf{7} & \mathbf{4}+\mathbf{6}+\mathbf{7} & \mathbf{2}+\mathbf{3}+\mathbf{5}+\mathbf{6}+\mathbf{7} & \mathbf{1}+\mathbf{4}+\mathbf{5}+\mathbf{6}+\mathbf{7} & \mathbf{3}+\mathbf{4}+\mathbf{5}+\mathbf{6}+\mathbf{7} \\ \mathbf{7} & \mathbf{7} & \mathbf{5}+\mathbf{6}+\mathbf{7} & \mathbf{5}+\mathbf{6}+\mathbf{7} & \mathbf{3}+\mathbf{4}+\mathbf{5}+\mathbf{6}+\mathbf{7} & \mathbf{3}+\mathbf{4}+\mathbf{5}+\mathbf{6}+\mathbf{7} & \mathbf{1}+\mathbf{2}+\mathbf{3}+\mathbf{4}+\mathbf{5}+\mathbf{6}+\mathbf{7} \\ \hline \end{array}\]The following elements form non-trivial sub fusion rings
| Elements | SubRing |
|---|---|
| \(\{\mathbf{1},\mathbf{2}\}\) | \(\mathbb{Z}_2:\ \text{FR}^{2,0}_{1}\) |
Frobenius-Perron Dimensions
| Particle | Numeric | Symbolic |
|---|---|---|
| \(\mathbf{1}\) | \(1.\) | \(1\) |
| \(\mathbf{2}\) | \(1.\) | \(1\) |
| \(\mathbf{3}\) | \(2.80194\) | \(\text{Root}\left[x^3-4 x^2+3 x+1,3\right]\) |
| \(\mathbf{4}\) | \(2.80194\) | \(\text{Root}\left[x^3-4 x^2+3 x+1,3\right]\) |
| \(\mathbf{5}\) | \(4.04892\) | \(\text{Root}\left[x^3-3 x^2-4 x-1,3\right]\) |
| \(\mathbf{6}\) | \(4.04892\) | \(\text{Root}\left[x^3-3 x^2-4 x-1,3\right]\) |
| \(\mathbf{7}\) | \(4.49396\) | \(\text{Root}\left[x^3-4 x^2-4 x+8,3\right]\) |
| \(\mathcal{D}_{FP}^2\) | \(70.6848\) | \(2 \text{Root}\left[x^3-4 x^2+3 x+1,3\right]^2+2 \text{Root}\left[x^3-3 x^2-4 x-1,3\right]^2+\text{Root}\left[x^3-4 x^2-4 x+8,3\right]^2+2\) |
Characters
The symbolic character table is the following
\[\begin{array}{|ccccccc|} \hline \mathbf{1} & \mathbf{2} & \mathbf{3} & \mathbf{4} & \mathbf{6} & \mathbf{5} & \mathbf{7} \\ \hline 1 & 1 & \text{Root}\left[x^3-4 x^2+3 x+1,3\right] & \text{Root}\left[x^3-4 x^2+3 x+1,3\right] & \text{Root}\left[x^3-3 x^2-4 x-1,3\right] & \text{Root}\left[x^3-3 x^2-4 x-1,3\right] & \text{Root}\left[x^3-4 x^2-4 x+8,3\right] \\ 1 & 1 & \text{Root}\left[x^3-4 x^2+3 x+1,2\right] & \text{Root}\left[x^3-4 x^2+3 x+1,2\right] & \text{Root}\left[x^3-3 x^2-4 x-1,2\right] & \text{Root}\left[x^3-3 x^2-4 x-1,2\right] & \text{Root}\left[x^3-4 x^2-4 x+8,1\right] \\ 1 & 1 & \text{Root}\left[x^3-4 x^2+3 x+1,1\right] & \text{Root}\left[x^3-4 x^2+3 x+1,1\right] & \text{Root}\left[x^3-3 x^2-4 x-1,1\right] & \text{Root}\left[x^3-3 x^2-4 x-1,1\right] & \text{Root}\left[x^3-4 x^2-4 x+8,2\right] \\ 1 & 1 & -1 & -1 & 1 & 1 & -1 \\ 1 & -1 & \text{Root}\left[x^3+2 x^2-x-1,1\right] & \text{Root}\left[x^3-2 x^2-x+1,3\right] & \text{Root}\left[x^3-x^2-2 x+1,3\right] & \text{Root}\left[x^3+x^2-2 x-1,1\right] & 0 \\ 1 & -1 & \text{Root}\left[x^3+2 x^2-x-1,3\right] & \text{Root}\left[x^3-2 x^2-x+1,1\right] & \text{Root}\left[x^3-x^2-2 x+1,2\right] & \text{Root}\left[x^3+x^2-2 x-1,2\right] & 0 \\ 1 & -1 & \text{Root}\left[x^3+2 x^2-x-1,2\right] & \text{Root}\left[x^3-2 x^2-x+1,2\right] & \text{Root}\left[x^3-x^2-2 x+1,1\right] & \text{Root}\left[x^3+x^2-2 x-1,3\right] & 0 \\ \hline \end{array}\]The numeric character table is the following
\[\begin{array}{|rrrrrrr|} \hline \mathbf{1} & \mathbf{2} & \mathbf{3} & \mathbf{4} & \mathbf{6} & \mathbf{5} & \mathbf{7} \\ \hline 1.000 & 1.000 & 2.802 & 2.802 & 4.049 & 4.049 & 4.494 \\ 1.000 & 1.000 & 1.445 & 1.445 & -0.3569 & -0.3569 & -1.604 \\ 1.000 & 1.000 & -0.2470 & -0.2470 & -0.6920 & -0.6920 & 1.110 \\ 1.000 & 1.000 & -1.000 & -1.000 & 1.000 & 1.000 & -1.000 \\ 1.000 & -1.000 & -2.247 & 2.247 & 1.802 & -1.802 & 0 \\ 1.000 & -1.000 & 0.8019 & -0.8019 & 0.4450 & -0.4450 & 0 \\ 1.000 & -1.000 & -0.5550 & 0.5550 & -1.247 & 1.247 & 0 \\ \hline \end{array}\]Representations of $SL_2(\mathbb{Z})$
This fusion ring does not provide any representations of $SL_2(\mathbb{Z}).$
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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