\(\text{PSU}(2)_{14}:\ \text{FR}^{8,0}_{31}\)
Fusion Rules
\[\begin{array}{|llllllll|} \hline \mathbf{1} & \mathbf{2} & \mathbf{3} & \mathbf{4} & \mathbf{5} & \mathbf{6} & \mathbf{7} & \mathbf{8} \\ \mathbf{2} & \mathbf{1} & \mathbf{4} & \mathbf{3} & \mathbf{6} & \mathbf{5} & \mathbf{8} & \mathbf{7} \\ \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{7}+\mathbf{8} & \mathbf{5}+\mathbf{7}+\mathbf{8} \\ \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{8} & \mathbf{6}+\mathbf{7}+\mathbf{8} \\ \mathbf{5} & \mathbf{6} & \mathbf{4}+\mathbf{6}+\mathbf{8} & \mathbf{3}+\mathbf{5}+\mathbf{7} & \mathbf{1}+\mathbf{4}+\mathbf{6}+\mathbf{7}+\mathbf{8} & \mathbf{2}+\mathbf{3}+\mathbf{5}+\mathbf{7}+\mathbf{8} & \mathbf{4}+\mathbf{5}+\mathbf{6}+\mathbf{7}+\mathbf{8} & \mathbf{3}+\mathbf{5}+\mathbf{6}+\mathbf{7}+\mathbf{8} \\ \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{8} & \mathbf{1}+\mathbf{4}+\mathbf{6}+\mathbf{7}+\mathbf{8} & \mathbf{3}+\mathbf{5}+\mathbf{6}+\mathbf{7}+\mathbf{8} & \mathbf{4}+\mathbf{5}+\mathbf{6}+\mathbf{7}+\mathbf{8} \\ \mathbf{7} & \mathbf{8} & \mathbf{6}+\mathbf{7}+\mathbf{8} & \mathbf{5}+\mathbf{7}+\mathbf{8} & \mathbf{4}+\mathbf{5}+\mathbf{6}+\mathbf{7}+\mathbf{8} & \mathbf{3}+\mathbf{5}+\mathbf{6}+\mathbf{7}+\mathbf{8} & \mathbf{1}+\mathbf{3}+\mathbf{4}+\mathbf{5}+\mathbf{6}+\mathbf{7}+\mathbf{8} & \mathbf{2}+\mathbf{3}+\mathbf{4}+\mathbf{5}+\mathbf{6}+\mathbf{7}+\mathbf{8} \\ \mathbf{8} & \mathbf{7} & \mathbf{5}+\mathbf{7}+\mathbf{8} & \mathbf{6}+\mathbf{7}+\mathbf{8} & \mathbf{3}+\mathbf{5}+\mathbf{6}+\mathbf{7}+\mathbf{8} & \mathbf{4}+\mathbf{5}+\mathbf{6}+\mathbf{7}+\mathbf{8} & \mathbf{2}+\mathbf{3}+\mathbf{4}+\mathbf{5}+\mathbf{6}+\mathbf{7}+\mathbf{8} & \mathbf{1}+\mathbf{3}+\mathbf{4}+\mathbf{5}+\mathbf{6}+\mathbf{7}+\mathbf{8} \\ \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 |
|---|---|---|
| \(\mathbf{1}\) | \(1.\) | \(1\) |
| \(\mathbf{2}\) | \(1.\) | \(1\) |
| \(\mathbf{3}\) | \(2.84776\) | \(1+\sqrt{2+\sqrt{2}}\) |
| \(\mathbf{4}\) | \(2.84776\) | \(1+\sqrt{2+\sqrt{2}}\) |
| \(\mathbf{5}\) | \(4.26197\) | \(\text{Root}\left[x^4-4 x^3-2 x^2+4 x-1,4\right]\) |
| \(\mathbf{6}\) | \(4.26197\) | \(\text{Root}\left[x^4-4 x^3-2 x^2+4 x-1,4\right]\) |
| \(\mathbf{7}\) | \(5.02734\) | \(\text{Root}\left[x^4-4 x^3-6 x^2+4 x+1,4\right]\) |
| \(\mathbf{8}\) | \(5.02734\) | \(\text{Root}\left[x^4-4 x^3-6 x^2+4 x+1,4\right]\) |
| \(\mathcal{D}_{FP}^2\) | \(105.097\) | \(2 \text{Root}\left[x^4-4 x^3-2 x^2+4 x-1,4\right]^2+2 \text{Root}\left[x^4-4 x^3-6 x^2+4 x+1,4\right]^2+2+2 \left(1+\sqrt{2+\sqrt{2}}\right)^2\) |
Characters
The symbolic character table is the following
\[\begin{array}{|cccccccc|} \hline \mathbf{1} & \mathbf{2} & \mathbf{3} & \mathbf{4} & \mathbf{6} & \mathbf{5} & \mathbf{8} & \mathbf{7} \\ \hline 1 & 1 & \text{Root}\left[x^4-4 x^3+2 x^2+4 x-1,4\right] & \text{Root}\left[x^4-4 x^3+2 x^2+4 x-1,4\right] & \text{Root}\left[x^4-4 x^3-2 x^2+4 x-1,4\right] & \text{Root}\left[x^4-4 x^3-2 x^2+4 x-1,4\right] & \text{Root}\left[x^4-4 x^3-6 x^2+4 x+1,4\right] & \text{Root}\left[x^4-4 x^3-6 x^2+4 x+1,4\right] \\ 1 & 1 & \text{Root}\left[x^4-4 x^3+2 x^2+4 x-1,3\right] & \text{Root}\left[x^4-4 x^3+2 x^2+4 x-1,3\right] & \text{Root}\left[x^4-4 x^3-2 x^2+4 x-1,2\right] & \text{Root}\left[x^4-4 x^3-2 x^2+4 x-1,2\right] & \text{Root}\left[x^4-4 x^3-6 x^2+4 x+1,1\right] & \text{Root}\left[x^4-4 x^3-6 x^2+4 x+1,1\right] \\ 1 & 1 & \text{Root}\left[x^4-4 x^3+2 x^2+4 x-1,2\right] & \text{Root}\left[x^4-4 x^3+2 x^2+4 x-1,2\right] & \text{Root}\left[x^4-4 x^3-2 x^2+4 x-1,1\right] & \text{Root}\left[x^4-4 x^3-2 x^2+4 x-1,1\right] & \text{Root}\left[x^4-4 x^3-6 x^2+4 x+1,3\right] & \text{Root}\left[x^4-4 x^3-6 x^2+4 x+1,3\right] \\ 1 & 1 & \text{Root}\left[x^4-4 x^3+2 x^2+4 x-1,1\right] & \text{Root}\left[x^4-4 x^3+2 x^2+4 x-1,1\right] & \text{Root}\left[x^4-4 x^3-2 x^2+4 x-1,3\right] & \text{Root}\left[x^4-4 x^3-2 x^2+4 x-1,3\right] & \text{Root}\left[x^4-4 x^3-6 x^2+4 x+1,2\right] & \text{Root}\left[x^4-4 x^3-6 x^2+4 x+1,2\right] \\ 1 & -1 & -1-\sqrt{2} & 1+\sqrt{2} & 1+\sqrt{2} & -1-\sqrt{2} & 1 & -1 \\ 1 & -1 & \sqrt{2}-1 & 1-\sqrt{2} & 1-\sqrt{2} & \sqrt{2}-1 & 1 & -1 \\ 1 & -1 & 1 & -1 & 1 & -1 & -1 & 1 \\ 1 & -1 & -1 & 1 & -1 & 1 & -1 & 1 \\ \hline \end{array}\]The numeric character table is the following
\[\begin{array}{|rrrrrrrr|} \hline \mathbf{1} & \mathbf{2} & \mathbf{3} & \mathbf{4} & \mathbf{6} & \mathbf{5} & \mathbf{8} & \mathbf{7} \\ \hline 1.000 & 1.000 & 2.848 & 2.848 & 4.262 & 4.262 & 5.027 & 5.027 \\ 1.000 & 1.000 & 1.765 & 1.765 & 0.3512 & 0.3512 & -1.497 & -1.497 \\ 1.000 & 1.000 & 0.2346 & 0.2346 & -1.180 & -1.180 & 0.6682 & 0.6682 \\ 1.000 & 1.000 & -0.8478 & -0.8478 & 0.5665 & 0.5665 & -0.1989 & -0.1989 \\ 1.000 & -1.000 & -2.414 & 2.414 & 2.414 & -2.414 & 1.000 & -1.000 \\ 1.000 & -1.000 & 0.4142 & -0.4142 & -0.4142 & 0.4142 & 1.000 & -1.000 \\ 1.000 & -1.000 & 1.000 & -1.000 & 1.000 & -1.000 & -1.000 & 1.000 \\ 1.000 & -1.000 & -1.000 & 1.000 & -1.000 & 1.000 & -1.000 & 1.000 \\ \hline \end{array}\]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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