\(\text{Fib$\times $Potts}:\ \text{FR}^{8,4}_{9}\)
Fusion Rules
\[\begin{array}{|llllllll|} \hline \mathbf{1} & \mathbf{2} & \mathbf{3} & \mathbf{4} & \mathbf{5} & \mathbf{6} & \mathbf{7} & \mathbf{8} \\ \mathbf{2} & \mathbf{3} & \mathbf{1} & \mathbf{6} & \mathbf{4} & \mathbf{5} & \mathbf{7} & \mathbf{8} \\ \mathbf{3} & \mathbf{1} & \mathbf{2} & \mathbf{5} & \mathbf{6} & \mathbf{4} & \mathbf{7} & \mathbf{8} \\ \mathbf{4} & \mathbf{6} & \mathbf{5} & \mathbf{1}+\mathbf{4} & \mathbf{3}+\mathbf{5} & \mathbf{2}+\mathbf{6} & \mathbf{8} & \mathbf{7}+\mathbf{8} \\ \mathbf{5} & \mathbf{4} & \mathbf{6} & \mathbf{3}+\mathbf{5} & \mathbf{2}+\mathbf{6} & \mathbf{1}+\mathbf{4} & \mathbf{8} & \mathbf{7}+\mathbf{8} \\ \mathbf{6} & \mathbf{5} & \mathbf{4} & \mathbf{2}+\mathbf{6} & \mathbf{1}+\mathbf{4} & \mathbf{3}+\mathbf{5} & \mathbf{8} & \mathbf{7}+\mathbf{8} \\ \mathbf{7} & \mathbf{7} & \mathbf{7} & \mathbf{8} & \mathbf{8} & \mathbf{8} & \mathbf{1}+\mathbf{2}+\mathbf{3} & \mathbf{4}+\mathbf{5}+\mathbf{6} \\ \mathbf{8} & \mathbf{8} & \mathbf{8} & \mathbf{7}+\mathbf{8} & \mathbf{7}+\mathbf{8} & \mathbf{7}+\mathbf{8} & \mathbf{4}+\mathbf{5}+\mathbf{6} & \mathbf{1}+\mathbf{2}+\mathbf{3}+\mathbf{4}+\mathbf{5}+\mathbf{6} \\ \hline \end{array}\]The fusion rules are invariant under the group generated by the following permutations:
\[\{(\mathbf{2} \ \mathbf{3}) (\mathbf{5} \ \mathbf{6})\}\]The following particles form non-trivial sub fusion rings
| Particles | SubRing |
|---|---|
| \(\{\mathbf{1},\mathbf{4}\}\) | \(\text{Fib}:\ \text{FR}^{2,0}_{2}\) |
| \(\{\mathbf{1},\mathbf{2},\mathbf{3}\}\) | \(\mathbb{Z}_3:\ \text{FR}^{3,2}_{1}\) |
| \(\{\mathbf{1},\mathbf{2},\mathbf{3},\mathbf{7}\}\) | \(\text{Potts}:\ \text{FR}^{4,2}_{2}\) |
| \(\{\mathbf{1},\mathbf{2},\mathbf{3},\mathbf{4},\mathbf{5},\mathbf{6}\}\) | \(\text{Fib$\times $}\mathbb{Z}_3:\ \text{FR}^{6,4}_{5}\) |
Quantum Dimensions
| Particle | Numeric | Symbolic |
|---|---|---|
| \(\mathbf{1}\) | \(1.\) | \(1\) |
| \(\mathbf{2}\) | \(1.\) | \(1\) |
| \(\mathbf{3}\) | \(1.\) | \(1\) |
| \(\mathbf{4}\) | \(1.61803\) | \(\frac{1}{2} \left(1+\sqrt{5}\right)\) |
| \(\mathbf{5}\) | \(1.61803\) | \(\frac{1}{2} \left(1+\sqrt{5}\right)\) |
| \(\mathbf{6}\) | \(1.61803\) | \(\frac{1}{2} \left(1+\sqrt{5}\right)\) |
| \(\mathbf{7}\) | \(1.73205\) | \(\sqrt{3}\) |
| \(\mathbf{8}\) | \(2.80252\) | \(\sqrt{\frac{3}{2} \left(3+\sqrt{5}\right)}\) |
| \(\mathcal{D}_{FP}^2\) | \(21.7082\) | \(6+\frac{3}{4} \left(1+\sqrt{5}\right)^2+\frac{3}{2} \left(3+\sqrt{5}\right)\) |
Characters
The symbolic character table is the following
\[\begin{array}{|cccccccc|} \hline \mathbf{1} & \mathbf{2} & \mathbf{3} & \mathbf{6} & \mathbf{5} & \mathbf{4} & \mathbf{7} & \mathbf{8} \\ \hline 1 & 1 & 1 & \frac{1}{2} \left(1+\sqrt{5}\right) & \frac{1}{2} \left(1+\sqrt{5}\right) & \frac{1}{2} \left(1+\sqrt{5}\right) & \sqrt{3} & \text{Root}\left[x^4-9 x^2+9,4\right] \\ 1 & 1 & 1 & \frac{1}{2} \left(1+\sqrt{5}\right) & \frac{1}{2} \left(1+\sqrt{5}\right) & \frac{1}{2} \left(1+\sqrt{5}\right) & -\sqrt{3} & \text{Root}\left[x^4-9 x^2+9,1\right] \\ 1 & 1 & 1 & \frac{1}{2} \left(1-\sqrt{5}\right) & \frac{1}{2} \left(1-\sqrt{5}\right) & \frac{1}{2} \left(1-\sqrt{5}\right) & \sqrt{3} & \text{Root}\left[x^4-9 x^2+9,2\right] \\ 1 & 1 & 1 & \frac{1}{2} \left(1-\sqrt{5}\right) & \frac{1}{2} \left(1-\sqrt{5}\right) & \frac{1}{2} \left(1-\sqrt{5}\right) & -\sqrt{3} & \text{Root}\left[x^4-9 x^2+9,3\right] \\ 1 & \frac{1}{2} \left(-1-i \sqrt{3}\right) & \frac{1}{2} \left(-1+i \sqrt{3}\right) & \text{Root}\left[x^4+x^3+2 x^2-x+1,4\right] & \text{Root}\left[x^4+x^3+2 x^2-x+1,3\right] & \frac{1}{2} \left(1-\sqrt{5}\right) & 0 & 0 \\ 1 & \frac{1}{2} \left(-1+i \sqrt{3}\right) & \frac{1}{2} \left(-1-i \sqrt{3}\right) & \text{Root}\left[x^4+x^3+2 x^2-x+1,3\right] & \text{Root}\left[x^4+x^3+2 x^2-x+1,4\right] & \frac{1}{2} \left(1-\sqrt{5}\right) & 0 & 0 \\ 1 & \frac{1}{2} \left(-1+i \sqrt{3}\right) & \frac{1}{2} \left(-1-i \sqrt{3}\right) & \text{Root}\left[x^4+x^3+2 x^2-x+1,2\right] & \text{Root}\left[x^4+x^3+2 x^2-x+1,1\right] & \frac{1}{2} \left(1+\sqrt{5}\right) & 0 & 0 \\ 1 & \frac{1}{2} \left(-1-i \sqrt{3}\right) & \frac{1}{2} \left(-1+i \sqrt{3}\right) & \text{Root}\left[x^4+x^3+2 x^2-x+1,1\right] & \text{Root}\left[x^4+x^3+2 x^2-x+1,2\right] & \frac{1}{2} \left(1+\sqrt{5}\right) & 0 & 0 \\ \hline \end{array}\]The numeric character table is the following
\[\begin{array}{|rrrrrrrr|} \hline \mathbf{1} & \mathbf{2} & \mathbf{3} & \mathbf{6} & \mathbf{5} & \mathbf{4} & \mathbf{7} & \mathbf{8} \\ \hline 1.000 & 1.000 & 1.000 & 1.618 & 1.618 & 1.618 & 1.732 & 2.803 \\ 1.000 & 1.000 & 1.000 & 1.618 & 1.618 & 1.618 & -1.732 & -2.803 \\ 1.000 & 1.000 & 1.000 & -0.6180 & -0.6180 & -0.6180 & 1.732 & -1.070 \\ 1.000 & 1.000 & 1.000 & -0.6180 & -0.6180 & -0.6180 & -1.732 & 1.070 \\ 1.000 & -0.5000-0.8660 i & -0.5000+0.8660 i & 0.3090+0.5352 i & 0.3090-0.5352 i & -0.6180 & 0 & 0 \\ 1.000 & -0.5000+0.8660 i & -0.5000-0.8660 i & 0.3090-0.5352 i & 0.3090+0.5352 i & -0.6180 & 0 & 0 \\ 1.000 & -0.5000+0.8660 i & -0.5000-0.8660 i & -0.809+1.401 i & -0.809-1.401 i & 1.618 & 0 & 0 \\ 1.000 & -0.5000-0.8660 i & -0.5000+0.8660 i & -0.809-1.401 i & -0.809+1.401 i & 1.618 & 0 & 0 \\ \hline \end{array}\]Modular Data
This fusion ring does not have any matching \(S\)-and \(T\)-matrices.
Adjoint Subring
Particles \(\mathbf{1}, \mathbf{2}, \mathbf{3}, \mathbf{4}, \mathbf{5}, \mathbf{6}\), form the adjoint subring \(\text{Fib$\times $}\mathbb{Z}_3:\ \text{FR}^{6,4}_{5}\) .
The upper central series is the following: \(\text{Fib$\times $Potts} \underset{ \mathbf{1}, \mathbf{2}, \mathbf{3}, \mathbf{4}, \mathbf{5}, \mathbf{6} }{\supset} \text{Fib$\times $}\mathbb{Z}_3 \underset{ \mathbf{1}, \mathbf{2} }{\supset} \text{Fib}\)
Universal grading
Each particle can be graded as follows: \(\text{deg}(\mathbf{1}) = \mathbf{1}', \text{deg}(\mathbf{2}) = \mathbf{1}', \text{deg}(\mathbf{3}) = \mathbf{1}', \text{deg}(\mathbf{4}) = \mathbf{1}', \text{deg}(\mathbf{5}) = \mathbf{1}', \text{deg}(\mathbf{6}) = \mathbf{1}', \text{deg}(\mathbf{7}) = \mathbf{2}', \text{deg}(\mathbf{8}) = \mathbf{2}'\), where the degrees form the group \(\mathbb{Z}_2\) with multiplication table:
\[\begin{array}{|ll|} \hline \mathbf{1}' & \mathbf{2}' \\ \mathbf{2}' & \mathbf{1}' \\ \hline \end{array}\]Categorifications
Data
Download links for numeric data: