\(\text{Fib$\times $}\text{PSU}(2)_6:\ \text{FR}^{8,0}_{16}\)
Fusion Rules
\[\begin{array}{|llllllll|} \hline \mathbf{1} & \mathbf{2} & \mathbf{3} & \mathbf{4} & \mathbf{5} & \mathbf{6} & \mathbf{7} & \mathbf{8} \\ \mathbf{2} & \mathbf{1} & \mathbf{4} & \mathbf{3} & \mathbf{6} & \mathbf{5} & \mathbf{8} & \mathbf{7} \\ \mathbf{3} & \mathbf{4} & \mathbf{1}+\mathbf{4} & \mathbf{2}+\mathbf{3} & \mathbf{8} & \mathbf{7} & \mathbf{6}+\mathbf{8} & \mathbf{5}+\mathbf{7} \\ \mathbf{4} & \mathbf{3} & \mathbf{2}+\mathbf{3} & \mathbf{1}+\mathbf{4} & \mathbf{7} & \mathbf{8} & \mathbf{5}+\mathbf{7} & \mathbf{6}+\mathbf{8} \\ \mathbf{5} & \mathbf{6} & \mathbf{8} & \mathbf{7} & \mathbf{1}+\mathbf{5}+\mathbf{6} & \mathbf{2}+\mathbf{5}+\mathbf{6} & \mathbf{4}+\mathbf{7}+\mathbf{8} & \mathbf{3}+\mathbf{7}+\mathbf{8} \\ \mathbf{6} & \mathbf{5} & \mathbf{7} & \mathbf{8} & \mathbf{2}+\mathbf{5}+\mathbf{6} & \mathbf{1}+\mathbf{5}+\mathbf{6} & \mathbf{3}+\mathbf{7}+\mathbf{8} & \mathbf{4}+\mathbf{7}+\mathbf{8} \\ \mathbf{7} & \mathbf{8} & \mathbf{6}+\mathbf{8} & \mathbf{5}+\mathbf{7} & \mathbf{4}+\mathbf{7}+\mathbf{8} & \mathbf{3}+\mathbf{7}+\mathbf{8} & \mathbf{1}+\mathbf{4}+\mathbf{5}+\mathbf{6}+\mathbf{7}+\mathbf{8} & \mathbf{2}+\mathbf{3}+\mathbf{5}+\mathbf{6}+\mathbf{7}+\mathbf{8} \\ \mathbf{8} & \mathbf{7} & \mathbf{5}+\mathbf{7} & \mathbf{6}+\mathbf{8} & \mathbf{3}+\mathbf{7}+\mathbf{8} & \mathbf{4}+\mathbf{7}+\mathbf{8} & \mathbf{2}+\mathbf{3}+\mathbf{5}+\mathbf{6}+\mathbf{7}+\mathbf{8} & \mathbf{1}+\mathbf{4}+\mathbf{5}+\mathbf{6}+\mathbf{7}+\mathbf{8} \\ \hline \end{array}\]The fusion rules are invariant under the group generated by the following permutations:
\[\{(\mathbf{5} \ \mathbf{6}) (\mathbf{7} \ \mathbf{8})\}\]The following particles form non-trivial sub fusion rings
| Particles | SubRing |
|---|---|
| \(\{\mathbf{1},\mathbf{2}\}\) | \(\mathbb{Z}_2:\ \text{FR}^{2,0}_{1}\) |
| \(\{\mathbf{1},\mathbf{4}\}\) | \(\text{Fib}:\ \text{FR}^{2,0}_{2}\) |
| \(\{\mathbf{1},\mathbf{2},\mathbf{5},\mathbf{6}\}\) | \(\text{PSU(2})_6:\ \text{FR}^{4,0}_{4}\) |
| \(\{\mathbf{1},\mathbf{2},\mathbf{3},\mathbf{4}\}\) | \(\text{SU(2})_3:\ \text{FR}^{4,0}_{2}\) |
Quantum Dimensions
| Particle | Numeric | Symbolic |
|---|---|---|
| \(\mathbf{1}\) | \(1.\) | \(1\) |
| \(\mathbf{2}\) | \(1.\) | \(1\) |
| \(\mathbf{3}\) | \(1.61803\) | \(\frac{1}{2} \left(1+\sqrt{5}\right)\) |
| \(\mathbf{4}\) | \(1.61803\) | \(\frac{1}{2} \left(1+\sqrt{5}\right)\) |
| \(\mathbf{5}\) | \(2.41421\) | \(1+\sqrt{2}\) |
| \(\mathbf{6}\) | \(2.41421\) | \(1+\sqrt{2}\) |
| \(\mathbf{7}\) | \(3.90628\) | \(\text{Root}\left[x^4-2 x^3-7 x^2-2 x+1,4\right]\) |
| \(\mathbf{8}\) | \(3.90628\) | \(\text{Root}\left[x^4-2 x^3-7 x^2-2 x+1,4\right]\) |
| \(\mathcal{D}_{FP}^2\) | \(49.411\) | \(2 \text{Root}\left[x^4-2 x^3-7 x^2-2 x+1,4\right]^2+2+2 \left(1+\sqrt{2}\right)^2+\frac{1}{2} \left(1+\sqrt{5}\right)^2\) |
Characters
The symbolic character table is the following
\[\begin{array}{|cccccccc|} \hline \mathbf{1} & \mathbf{2} & \mathbf{4} & \mathbf{3} & \mathbf{6} & \mathbf{5} & \mathbf{7} & \mathbf{8} \\ \hline 1 & 1 & \frac{1}{2} \left(1+\sqrt{5}\right) & \frac{1}{2} \left(1+\sqrt{5}\right) & 1+\sqrt{2} & 1+\sqrt{2} & \text{Root}\left[x^4-2 x^3-7 x^2-2 x+1,4\right] & \text{Root}\left[x^4-2 x^3-7 x^2-2 x+1,4\right] \\ 1 & 1 & \frac{1}{2} \left(1+\sqrt{5}\right) & \frac{1}{2} \left(1+\sqrt{5}\right) & 1-\sqrt{2} & 1-\sqrt{2} & \text{Root}\left[x^4-2 x^3-7 x^2-2 x+1,2\right] & \text{Root}\left[x^4-2 x^3-7 x^2-2 x+1,2\right] \\ 1 & 1 & \frac{1}{2} \left(1-\sqrt{5}\right) & \frac{1}{2} \left(1-\sqrt{5}\right) & 1+\sqrt{2} & 1+\sqrt{2} & \text{Root}\left[x^4-2 x^3-7 x^2-2 x+1,1\right] & \text{Root}\left[x^4-2 x^3-7 x^2-2 x+1,1\right] \\ 1 & 1 & \frac{1}{2} \left(1-\sqrt{5}\right) & \frac{1}{2} \left(1-\sqrt{5}\right) & 1-\sqrt{2} & 1-\sqrt{2} & \text{Root}\left[x^4-2 x^3-7 x^2-2 x+1,3\right] & \text{Root}\left[x^4-2 x^3-7 x^2-2 x+1,3\right] \\ 1 & -1 & \frac{1}{2} \left(1+\sqrt{5}\right) & \frac{1}{2} \left(-1-\sqrt{5}\right) & 1 & -1 & \frac{1}{2} \left(-1-\sqrt{5}\right) & \frac{1}{2} \left(1+\sqrt{5}\right) \\ 1 & -1 & \frac{1}{2} \left(1+\sqrt{5}\right) & \frac{1}{2} \left(-1-\sqrt{5}\right) & -1 & 1 & \frac{1}{2} \left(1+\sqrt{5}\right) & \frac{1}{2} \left(-1-\sqrt{5}\right) \\ 1 & -1 & \frac{1}{2} \left(1-\sqrt{5}\right) & \frac{1}{2} \left(\sqrt{5}-1\right) & 1 & -1 & \frac{1}{2} \left(\sqrt{5}-1\right) & \frac{1}{2} \left(1-\sqrt{5}\right) \\ 1 & -1 & \frac{1}{2} \left(1-\sqrt{5}\right) & \frac{1}{2} \left(\sqrt{5}-1\right) & -1 & 1 & \frac{1}{2} \left(1-\sqrt{5}\right) & \frac{1}{2} \left(\sqrt{5}-1\right) \\ \hline \end{array}\]The numeric character table is the following
\[\begin{array}{|rrrrrrrr|} \hline \mathbf{1} & \mathbf{2} & \mathbf{4} & \mathbf{3} & \mathbf{6} & \mathbf{5} & \mathbf{7} & \mathbf{8} \\ \hline 1.000 & 1.000 & 1.618 & 1.618 & 2.414 & 2.414 & 3.906 & 3.906 \\ 1.000 & 1.000 & 1.618 & 1.618 & -0.4142 & -0.4142 & -0.6702 & -0.6702 \\ 1.000 & 1.000 & -0.6180 & -0.6180 & 2.414 & 2.414 & -1.492 & -1.492 \\ 1.000 & 1.000 & -0.6180 & -0.6180 & -0.4142 & -0.4142 & 0.2560 & 0.2560 \\ 1.000 & -1.000 & 1.618 & -1.618 & 1.000 & -1.000 & -1.618 & 1.618 \\ 1.000 & -1.000 & 1.618 & -1.618 & -1.000 & 1.000 & 1.618 & -1.618 \\ 1.000 & -1.000 & -0.6180 & 0.6180 & 1.000 & -1.000 & 0.6180 & -0.6180 \\ 1.000 & -1.000 & -0.6180 & 0.6180 & -1.000 & 1.000 & -0.6180 & 0.6180 \\ \hline \end{array}\]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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