\(\text{Fib$\times $Fib$\times $Fib}:\ \text{FR}^{8,0}_{22}\)
Fusion Rules
\[\begin{array}{|llllllll|} \hline \mathbf{1} & \mathbf{2} & \mathbf{3} & \mathbf{4} & \mathbf{5} & \mathbf{6} & \mathbf{7} & \mathbf{8} \\ \mathbf{2} & \mathbf{1}+\mathbf{2} & \mathbf{7} & \mathbf{5} & \mathbf{4}+\mathbf{5} & \mathbf{8} & \mathbf{3}+\mathbf{7} & \mathbf{6}+\mathbf{8} \\ \mathbf{3} & \mathbf{7} & \mathbf{1}+\mathbf{3} & \mathbf{6} & \mathbf{8} & \mathbf{4}+\mathbf{6} & \mathbf{2}+\mathbf{7} & \mathbf{5}+\mathbf{8} \\ \mathbf{4} & \mathbf{5} & \mathbf{6} & \mathbf{1}+\mathbf{4} & \mathbf{2}+\mathbf{5} & \mathbf{3}+\mathbf{6} & \mathbf{8} & \mathbf{7}+\mathbf{8} \\ \mathbf{5} & \mathbf{4}+\mathbf{5} & \mathbf{8} & \mathbf{2}+\mathbf{5} & \mathbf{1}+\mathbf{2}+\mathbf{4}+\mathbf{5} & \mathbf{7}+\mathbf{8} & \mathbf{6}+\mathbf{8} & \mathbf{3}+\mathbf{6}+\mathbf{7}+\mathbf{8} \\ \mathbf{6} & \mathbf{8} & \mathbf{4}+\mathbf{6} & \mathbf{3}+\mathbf{6} & \mathbf{7}+\mathbf{8} & \mathbf{1}+\mathbf{3}+\mathbf{4}+\mathbf{6} & \mathbf{5}+\mathbf{8} & \mathbf{2}+\mathbf{5}+\mathbf{7}+\mathbf{8} \\ \mathbf{7} & \mathbf{3}+\mathbf{7} & \mathbf{2}+\mathbf{7} & \mathbf{8} & \mathbf{6}+\mathbf{8} & \mathbf{5}+\mathbf{8} & \mathbf{1}+\mathbf{2}+\mathbf{3}+\mathbf{7} & \mathbf{4}+\mathbf{5}+\mathbf{6}+\mathbf{8} \\ \mathbf{8} & \mathbf{6}+\mathbf{8} & \mathbf{5}+\mathbf{8} & \mathbf{7}+\mathbf{8} & \mathbf{3}+\mathbf{6}+\mathbf{7}+\mathbf{8} & \mathbf{2}+\mathbf{5}+\mathbf{7}+\mathbf{8} & \mathbf{4}+\mathbf{5}+\mathbf{6}+\mathbf{8} & \mathbf{1}+\mathbf{2}+\mathbf{3}+\mathbf{4}+\mathbf{5}+\mathbf{6}+\mathbf{7}+\mathbf{8} \\ \hline \end{array}\]The fusion rules are invariant under the group generated by the following permutations:
\[\{(\mathbf{2} \ \mathbf{3}) (\mathbf{5} \ \mathbf{6}), (\mathbf{2} \ \mathbf{4}) (\mathbf{6} \ \mathbf{7}), (\mathbf{3} \ \mathbf{4}) (\mathbf{5} \ \mathbf{7})\}\]The following particles form non-trivial sub fusion rings
| Particles | SubRing |
|---|---|
| \(\{\mathbf{1},\mathbf{2}\}\) | \(\text{Fib}:\ \text{FR}^{2,0}_{2}\) |
| \(\{\mathbf{1},\mathbf{3}\}\) | \(\text{Fib}:\ \text{FR}^{2,0}_{2}\) |
| \(\{\mathbf{1},\mathbf{4}\}\) | \(\text{Fib}:\ \text{FR}^{2,0}_{2}\) |
| \(\{\mathbf{1},\mathbf{2},\mathbf{3},\mathbf{7}\}\) | \(\text{Fib$\times $Fib}:\ \text{FR}^{4,0}_{5}\) |
| \(\{\mathbf{1},\mathbf{2},\mathbf{4},\mathbf{5}\}\) | \(\text{Fib$\times $Fib}:\ \text{FR}^{4,0}_{5}\) |
| \(\{\mathbf{1},\mathbf{3},\mathbf{4},\mathbf{6}\}\) | \(\text{Fib$\times $Fib}:\ \text{FR}^{4,0}_{5}\) |
Quantum Dimensions
| Particle | Numeric | Symbolic |
|---|---|---|
| \(\mathbf{1}\) | \(1.\) | \(1\) |
| \(\mathbf{2}\) | \(1.61803\) | \(\frac{1}{2} \left(1+\sqrt{5}\right)\) |
| \(\mathbf{3}\) | \(1.61803\) | \(\frac{1}{2} \left(1+\sqrt{5}\right)\) |
| \(\mathbf{4}\) | \(1.61803\) | \(\frac{1}{2} \left(1+\sqrt{5}\right)\) |
| \(\mathbf{5}\) | \(2.61803\) | \(\frac{1}{2} \left(3+\sqrt{5}\right)\) |
| \(\mathbf{6}\) | \(2.61803\) | \(\frac{1}{2} \left(3+\sqrt{5}\right)\) |
| \(\mathbf{7}\) | \(2.61803\) | \(\frac{1}{2} \left(3+\sqrt{5}\right)\) |
| \(\mathbf{8}\) | \(4.23607\) | \(2+\sqrt{5}\) |
| \(\mathcal{D}_{FP}^2\) | \(47.3607\) | \(1+\frac{3}{4} \left(1+\sqrt{5}\right)^2+\left(2+\sqrt{5}\right)^2+\frac{3}{4} \left(3+\sqrt{5}\right)^2\) |
Characters
The symbolic character table is the following
\[\begin{array}{|cccccccc|} \hline \mathbf{1} & \mathbf{2} & \mathbf{3} & \mathbf{4} & \mathbf{5} & \mathbf{6} & \mathbf{7} & \mathbf{8} \\ \hline 1 & \frac{1}{2} \left(1+\sqrt{5}\right) & \frac{1}{2} \left(1+\sqrt{5}\right) & \frac{1}{2} \left(1+\sqrt{5}\right) & \frac{1}{2} \left(3+\sqrt{5}\right) & \frac{1}{2} \left(3+\sqrt{5}\right) & \frac{1}{2} \left(3+\sqrt{5}\right) & 2+\sqrt{5} \\ 1 & \frac{1}{2} \left(1-\sqrt{5}\right) & \frac{1}{2} \left(1+\sqrt{5}\right) & \frac{1}{2} \left(1+\sqrt{5}\right) & -1 & \frac{1}{2} \left(3+\sqrt{5}\right) & -1 & \frac{1}{2} \left(-1-\sqrt{5}\right) \\ 1 & \frac{1}{2} \left(1+\sqrt{5}\right) & \frac{1}{2} \left(1-\sqrt{5}\right) & \frac{1}{2} \left(1+\sqrt{5}\right) & \frac{1}{2} \left(3+\sqrt{5}\right) & -1 & -1 & \frac{1}{2} \left(-1-\sqrt{5}\right) \\ 1 & \frac{1}{2} \left(1+\sqrt{5}\right) & \frac{1}{2} \left(1+\sqrt{5}\right) & \frac{1}{2} \left(1-\sqrt{5}\right) & -1 & -1 & \frac{1}{2} \left(3+\sqrt{5}\right) & \frac{1}{2} \left(-1-\sqrt{5}\right) \\ 1 & \frac{1}{2} \left(1-\sqrt{5}\right) & \frac{1}{2} \left(1+\sqrt{5}\right) & \frac{1}{2} \left(1-\sqrt{5}\right) & \frac{1}{2} \left(3-\sqrt{5}\right) & -1 & -1 & \frac{1}{2} \left(\sqrt{5}-1\right) \\ 1 & \frac{1}{2} \left(1+\sqrt{5}\right) & \frac{1}{2} \left(1-\sqrt{5}\right) & \frac{1}{2} \left(1-\sqrt{5}\right) & -1 & \frac{1}{2} \left(3-\sqrt{5}\right) & -1 & \frac{1}{2} \left(\sqrt{5}-1\right) \\ 1 & \frac{1}{2} \left(1-\sqrt{5}\right) & \frac{1}{2} \left(1-\sqrt{5}\right) & \frac{1}{2} \left(1+\sqrt{5}\right) & -1 & -1 & \frac{1}{2} \left(3-\sqrt{5}\right) & \frac{1}{2} \left(\sqrt{5}-1\right) \\ 1 & \frac{1}{2} \left(1-\sqrt{5}\right) & \frac{1}{2} \left(1-\sqrt{5}\right) & \frac{1}{2} \left(1-\sqrt{5}\right) & \frac{1}{2} \left(3-\sqrt{5}\right) & \frac{1}{2} \left(3-\sqrt{5}\right) & \frac{1}{2} \left(3-\sqrt{5}\right) & 2-\sqrt{5} \\ \hline \end{array}\]The numeric character table is the following
\[\begin{array}{|rrrrrrrr|} \hline \mathbf{1} & \mathbf{2} & \mathbf{3} & \mathbf{4} & \mathbf{5} & \mathbf{6} & \mathbf{7} & \mathbf{8} \\ \hline 1.000 & 1.618 & 1.618 & 1.618 & 2.618 & 2.618 & 2.618 & 4.236 \\ 1.000 & -0.6180 & 1.618 & 1.618 & -1.000 & 2.618 & -1.000 & -1.618 \\ 1.000 & 1.618 & -0.6180 & 1.618 & 2.618 & -1.000 & -1.000 & -1.618 \\ 1.000 & 1.618 & 1.618 & -0.6180 & -1.000 & -1.000 & 2.618 & -1.618 \\ 1.000 & -0.6180 & 1.618 & -0.6180 & 0.3820 & -1.000 & -1.000 & 0.6180 \\ 1.000 & 1.618 & -0.6180 & -0.6180 & -1.000 & 0.3820 & -1.000 & 0.6180 \\ 1.000 & -0.6180 & -0.6180 & 1.618 & -1.000 & -1.000 & 0.3820 & 0.6180 \\ 1.000 & -0.6180 & -0.6180 & -0.6180 & 0.3820 & 0.3820 & 0.3820 & -0.2361 \\ \hline \end{array}\]Modular Data
The matching \(S\)-matrices and twist factors are the following
| \(S\)-matrix | Twist factors |
|---|---|
| \(\frac{1}{\sqrt{1+\frac{3}{4} \left(1+\sqrt{5}\right)^2+\left(2+\sqrt{5}\right)^2+\frac{3}{4} \left(3+\sqrt{5}\right)^2}}\left(\begin{array}{cccccccc} \sqrt{1+\frac{3}{4} \left(1+\sqrt{5}\right)^2+\left(2+\sqrt{5}\right)^2+\frac{3}{4} \left(3+\sqrt{5}\right)^2} \text{Root}\left[125 x^4-50 x^2+1,3\right] & \sqrt{1+\frac{3}{4} \left(1+\sqrt{5}\right)^2+\left(2+\sqrt{5}\right)^2+\frac{3}{4} \left(3+\sqrt{5}\right)^2} \text{Root}\left[125 x^4-25 x^2+1,3\right] & \sqrt{1+\frac{3}{4} \left(1+\sqrt{5}\right)^2+\left(2+\sqrt{5}\right)^2+\frac{3}{4} \left(3+\sqrt{5}\right)^2} \text{Root}\left[125 x^4-25 x^2+1,3\right] & \sqrt{1+\frac{3}{4} \left(1+\sqrt{5}\right)^2+\left(2+\sqrt{5}\right)^2+\frac{3}{4} \left(3+\sqrt{5}\right)^2} \text{Root}\left[125 x^4-25 x^2+1,3\right] & \sqrt{1+\frac{3}{4} \left(1+\sqrt{5}\right)^2+\left(2+\sqrt{5}\right)^2+\frac{3}{4} \left(3+\sqrt{5}\right)^2} \text{Root}\left[125 x^4-25 x^2+1,4\right] & \sqrt{1+\frac{3}{4} \left(1+\sqrt{5}\right)^2+\left(2+\sqrt{5}\right)^2+\frac{3}{4} \left(3+\sqrt{5}\right)^2} \text{Root}\left[125 x^4-25 x^2+1,4\right] & \sqrt{1+\frac{3}{4} \left(1+\sqrt{5}\right)^2+\left(2+\sqrt{5}\right)^2+\frac{3}{4} \left(3+\sqrt{5}\right)^2} \text{Root}\left[125 x^4-25 x^2+1,4\right] & \sqrt{1+\frac{3}{4} \left(1+\sqrt{5}\right)^2+\left(2+\sqrt{5}\right)^2+\frac{3}{4} \left(3+\sqrt{5}\right)^2} \text{Root}\left[125 x^4-50 x^2+1,4\right] \\ \sqrt{1+\frac{3}{4} \left(1+\sqrt{5}\right)^2+\left(2+\sqrt{5}\right)^2+\frac{3}{4} \left(3+\sqrt{5}\right)^2} \text{Root}\left[125 x^4-25 x^2+1,3\right] & \sqrt{1+\frac{3}{4} \left(1+\sqrt{5}\right)^2+\left(2+\sqrt{5}\right)^2+\frac{3}{4} \left(3+\sqrt{5}\right)^2} \text{Root}\left[125 x^4-50 x^2+1,2\right] & \sqrt{1+\frac{3}{4} \left(1+\sqrt{5}\right)^2+\left(2+\sqrt{5}\right)^2+\frac{3}{4} \left(3+\sqrt{5}\right)^2} \text{Root}\left[125 x^4-25 x^2+1,4\right] & \sqrt{1+\frac{3}{4} \left(1+\sqrt{5}\right)^2+\left(2+\sqrt{5}\right)^2+\frac{3}{4} \left(3+\sqrt{5}\right)^2} \text{Root}\left[125 x^4-25 x^2+1,4\right] & \sqrt{1+\frac{3}{4} \left(1+\sqrt{5}\right)^2+\left(2+\sqrt{5}\right)^2+\frac{3}{4} \left(3+\sqrt{5}\right)^2} \text{Root}\left[125 x^4-25 x^2+1,2\right] & \sqrt{1+\frac{3}{4} \left(1+\sqrt{5}\right)^2+\left(2+\sqrt{5}\right)^2+\frac{3}{4} \left(3+\sqrt{5}\right)^2} \text{Root}\left[125 x^4-50 x^2+1,4\right] & \sqrt{1+\frac{3}{4} \left(1+\sqrt{5}\right)^2+\left(2+\sqrt{5}\right)^2+\frac{3}{4} \left(3+\sqrt{5}\right)^2} \text{Root}\left[125 x^4-25 x^2+1,2\right] & \sqrt{1+\frac{3}{4} \left(1+\sqrt{5}\right)^2+\left(2+\sqrt{5}\right)^2+\frac{3}{4} \left(3+\sqrt{5}\right)^2} \text{Root}\left[125 x^4-25 x^2+1,1\right] \\ \sqrt{1+\frac{3}{4} \left(1+\sqrt{5}\right)^2+\left(2+\sqrt{5}\right)^2+\frac{3}{4} \left(3+\sqrt{5}\right)^2} \text{Root}\left[125 x^4-25 x^2+1,3\right] & \sqrt{1+\frac{3}{4} \left(1+\sqrt{5}\right)^2+\left(2+\sqrt{5}\right)^2+\frac{3}{4} \left(3+\sqrt{5}\right)^2} \text{Root}\left[125 x^4-25 x^2+1,4\right] & \sqrt{1+\frac{3}{4} \left(1+\sqrt{5}\right)^2+\left(2+\sqrt{5}\right)^2+\frac{3}{4} \left(3+\sqrt{5}\right)^2} \text{Root}\left[125 x^4-50 x^2+1,2\right] & \sqrt{1+\frac{3}{4} \left(1+\sqrt{5}\right)^2+\left(2+\sqrt{5}\right)^2+\frac{3}{4} \left(3+\sqrt{5}\right)^2} \text{Root}\left[125 x^4-25 x^2+1,4\right] & \sqrt{1+\frac{3}{4} \left(1+\sqrt{5}\right)^2+\left(2+\sqrt{5}\right)^2+\frac{3}{4} \left(3+\sqrt{5}\right)^2} \text{Root}\left[125 x^4-50 x^2+1,4\right] & \sqrt{1+\frac{3}{4} \left(1+\sqrt{5}\right)^2+\left(2+\sqrt{5}\right)^2+\frac{3}{4} \left(3+\sqrt{5}\right)^2} \text{Root}\left[125 x^4-25 x^2+1,2\right] & \sqrt{1+\frac{3}{4} \left(1+\sqrt{5}\right)^2+\left(2+\sqrt{5}\right)^2+\frac{3}{4} \left(3+\sqrt{5}\right)^2} \text{Root}\left[125 x^4-25 x^2+1,2\right] & \sqrt{1+\frac{3}{4} \left(1+\sqrt{5}\right)^2+\left(2+\sqrt{5}\right)^2+\frac{3}{4} \left(3+\sqrt{5}\right)^2} \text{Root}\left[125 x^4-25 x^2+1,1\right] \\ \sqrt{1+\frac{3}{4} \left(1+\sqrt{5}\right)^2+\left(2+\sqrt{5}\right)^2+\frac{3}{4} \left(3+\sqrt{5}\right)^2} \text{Root}\left[125 x^4-25 x^2+1,3\right] & \sqrt{1+\frac{3}{4} \left(1+\sqrt{5}\right)^2+\left(2+\sqrt{5}\right)^2+\frac{3}{4} \left(3+\sqrt{5}\right)^2} \text{Root}\left[125 x^4-25 x^2+1,4\right] & \sqrt{1+\frac{3}{4} \left(1+\sqrt{5}\right)^2+\left(2+\sqrt{5}\right)^2+\frac{3}{4} \left(3+\sqrt{5}\right)^2} \text{Root}\left[125 x^4-25 x^2+1,4\right] & \sqrt{1+\frac{3}{4} \left(1+\sqrt{5}\right)^2+\left(2+\sqrt{5}\right)^2+\frac{3}{4} \left(3+\sqrt{5}\right)^2} \text{Root}\left[125 x^4-50 x^2+1,2\right] & \sqrt{1+\frac{3}{4} \left(1+\sqrt{5}\right)^2+\left(2+\sqrt{5}\right)^2+\frac{3}{4} \left(3+\sqrt{5}\right)^2} \text{Root}\left[125 x^4-25 x^2+1,2\right] & \sqrt{1+\frac{3}{4} \left(1+\sqrt{5}\right)^2+\left(2+\sqrt{5}\right)^2+\frac{3}{4} \left(3+\sqrt{5}\right)^2} \text{Root}\left[125 x^4-25 x^2+1,2\right] & \sqrt{1+\frac{3}{4} \left(1+\sqrt{5}\right)^2+\left(2+\sqrt{5}\right)^2+\frac{3}{4} \left(3+\sqrt{5}\right)^2} \text{Root}\left[125 x^4-50 x^2+1,4\right] & \sqrt{1+\frac{3}{4} \left(1+\sqrt{5}\right)^2+\left(2+\sqrt{5}\right)^2+\frac{3}{4} \left(3+\sqrt{5}\right)^2} \text{Root}\left[125 x^4-25 x^2+1,1\right] \\ \sqrt{1+\frac{3}{4} \left(1+\sqrt{5}\right)^2+\left(2+\sqrt{5}\right)^2+\frac{3}{4} \left(3+\sqrt{5}\right)^2} \text{Root}\left[125 x^4-25 x^2+1,4\right] & \sqrt{1+\frac{3}{4} \left(1+\sqrt{5}\right)^2+\left(2+\sqrt{5}\right)^2+\frac{3}{4} \left(3+\sqrt{5}\right)^2} \text{Root}\left[125 x^4-25 x^2+1,2\right] & \sqrt{1+\frac{3}{4} \left(1+\sqrt{5}\right)^2+\left(2+\sqrt{5}\right)^2+\frac{3}{4} \left(3+\sqrt{5}\right)^2} \text{Root}\left[125 x^4-50 x^2+1,4\right] & \sqrt{1+\frac{3}{4} \left(1+\sqrt{5}\right)^2+\left(2+\sqrt{5}\right)^2+\frac{3}{4} \left(3+\sqrt{5}\right)^2} \text{Root}\left[125 x^4-25 x^2+1,2\right] & \sqrt{1+\frac{3}{4} \left(1+\sqrt{5}\right)^2+\left(2+\sqrt{5}\right)^2+\frac{3}{4} \left(3+\sqrt{5}\right)^2} \text{Root}\left[125 x^4-50 x^2+1,3\right] & \sqrt{1+\frac{3}{4} \left(1+\sqrt{5}\right)^2+\left(2+\sqrt{5}\right)^2+\frac{3}{4} \left(3+\sqrt{5}\right)^2} \text{Root}\left[125 x^4-25 x^2+1,1\right] & \sqrt{1+\frac{3}{4} \left(1+\sqrt{5}\right)^2+\left(2+\sqrt{5}\right)^2+\frac{3}{4} \left(3+\sqrt{5}\right)^2} \text{Root}\left[125 x^4-25 x^2+1,1\right] & \sqrt{1+\frac{3}{4} \left(1+\sqrt{5}\right)^2+\left(2+\sqrt{5}\right)^2+\frac{3}{4} \left(3+\sqrt{5}\right)^2} \text{Root}\left[125 x^4-25 x^2+1,3\right] \\ \sqrt{1+\frac{3}{4} \left(1+\sqrt{5}\right)^2+\left(2+\sqrt{5}\right)^2+\frac{3}{4} \left(3+\sqrt{5}\right)^2} \text{Root}\left[125 x^4-25 x^2+1,4\right] & \sqrt{1+\frac{3}{4} \left(1+\sqrt{5}\right)^2+\left(2+\sqrt{5}\right)^2+\frac{3}{4} \left(3+\sqrt{5}\right)^2} \text{Root}\left[125 x^4-50 x^2+1,4\right] & \sqrt{1+\frac{3}{4} \left(1+\sqrt{5}\right)^2+\left(2+\sqrt{5}\right)^2+\frac{3}{4} \left(3+\sqrt{5}\right)^2} \text{Root}\left[125 x^4-25 x^2+1,2\right] & \sqrt{1+\frac{3}{4} \left(1+\sqrt{5}\right)^2+\left(2+\sqrt{5}\right)^2+\frac{3}{4} \left(3+\sqrt{5}\right)^2} \text{Root}\left[125 x^4-25 x^2+1,2\right] & \sqrt{1+\frac{3}{4} \left(1+\sqrt{5}\right)^2+\left(2+\sqrt{5}\right)^2+\frac{3}{4} \left(3+\sqrt{5}\right)^2} \text{Root}\left[125 x^4-25 x^2+1,1\right] & \sqrt{1+\frac{3}{4} \left(1+\sqrt{5}\right)^2+\left(2+\sqrt{5}\right)^2+\frac{3}{4} \left(3+\sqrt{5}\right)^2} \text{Root}\left[125 x^4-50 x^2+1,3\right] & \sqrt{1+\frac{3}{4} \left(1+\sqrt{5}\right)^2+\left(2+\sqrt{5}\right)^2+\frac{3}{4} \left(3+\sqrt{5}\right)^2} \text{Root}\left[125 x^4-25 x^2+1,1\right] & \sqrt{1+\frac{3}{4} \left(1+\sqrt{5}\right)^2+\left(2+\sqrt{5}\right)^2+\frac{3}{4} \left(3+\sqrt{5}\right)^2} \text{Root}\left[125 x^4-25 x^2+1,3\right] \\ \sqrt{1+\frac{3}{4} \left(1+\sqrt{5}\right)^2+\left(2+\sqrt{5}\right)^2+\frac{3}{4} \left(3+\sqrt{5}\right)^2} \text{Root}\left[125 x^4-25 x^2+1,4\right] & \sqrt{1+\frac{3}{4} \left(1+\sqrt{5}\right)^2+\left(2+\sqrt{5}\right)^2+\frac{3}{4} \left(3+\sqrt{5}\right)^2} \text{Root}\left[125 x^4-25 x^2+1,2\right] & \sqrt{1+\frac{3}{4} \left(1+\sqrt{5}\right)^2+\left(2+\sqrt{5}\right)^2+\frac{3}{4} \left(3+\sqrt{5}\right)^2} \text{Root}\left[125 x^4-25 x^2+1,2\right] & \sqrt{1+\frac{3}{4} \left(1+\sqrt{5}\right)^2+\left(2+\sqrt{5}\right)^2+\frac{3}{4} \left(3+\sqrt{5}\right)^2} \text{Root}\left[125 x^4-50 x^2+1,4\right] & \sqrt{1+\frac{3}{4} \left(1+\sqrt{5}\right)^2+\left(2+\sqrt{5}\right)^2+\frac{3}{4} \left(3+\sqrt{5}\right)^2} \text{Root}\left[125 x^4-25 x^2+1,1\right] & \sqrt{1+\frac{3}{4} \left(1+\sqrt{5}\right)^2+\left(2+\sqrt{5}\right)^2+\frac{3}{4} \left(3+\sqrt{5}\right)^2} \text{Root}\left[125 x^4-25 x^2+1,1\right] & \sqrt{1+\frac{3}{4} \left(1+\sqrt{5}\right)^2+\left(2+\sqrt{5}\right)^2+\frac{3}{4} \left(3+\sqrt{5}\right)^2} \text{Root}\left[125 x^4-50 x^2+1,3\right] & \sqrt{1+\frac{3}{4} \left(1+\sqrt{5}\right)^2+\left(2+\sqrt{5}\right)^2+\frac{3}{4} \left(3+\sqrt{5}\right)^2} \text{Root}\left[125 x^4-25 x^2+1,3\right] \\ \sqrt{1+\frac{3}{4} \left(1+\sqrt{5}\right)^2+\left(2+\sqrt{5}\right)^2+\frac{3}{4} \left(3+\sqrt{5}\right)^2} \text{Root}\left[125 x^4-50 x^2+1,4\right] & \sqrt{1+\frac{3}{4} \left(1+\sqrt{5}\right)^2+\left(2+\sqrt{5}\right)^2+\frac{3}{4} \left(3+\sqrt{5}\right)^2} \text{Root}\left[125 x^4-25 x^2+1,1\right] & \sqrt{1+\frac{3}{4} \left(1+\sqrt{5}\right)^2+\left(2+\sqrt{5}\right)^2+\frac{3}{4} \left(3+\sqrt{5}\right)^2} \text{Root}\left[125 x^4-25 x^2+1,1\right] & \sqrt{1+\frac{3}{4} \left(1+\sqrt{5}\right)^2+\left(2+\sqrt{5}\right)^2+\frac{3}{4} \left(3+\sqrt{5}\right)^2} \text{Root}\left[125 x^4-25 x^2+1,1\right] & \sqrt{1+\frac{3}{4} \left(1+\sqrt{5}\right)^2+\left(2+\sqrt{5}\right)^2+\frac{3}{4} \left(3+\sqrt{5}\right)^2} \text{Root}\left[125 x^4-25 x^2+1,3\right] & \sqrt{1+\frac{3}{4} \left(1+\sqrt{5}\right)^2+\left(2+\sqrt{5}\right)^2+\frac{3}{4} \left(3+\sqrt{5}\right)^2} \text{Root}\left[125 x^4-25 x^2+1,3\right] & \sqrt{1+\frac{3}{4} \left(1+\sqrt{5}\right)^2+\left(2+\sqrt{5}\right)^2+\frac{3}{4} \left(3+\sqrt{5}\right)^2} \text{Root}\left[125 x^4-25 x^2+1,3\right] & \sqrt{1+\frac{3}{4} \left(1+\sqrt{5}\right)^2+\left(2+\sqrt{5}\right)^2+\frac{3}{4} \left(3+\sqrt{5}\right)^2} \text{Root}\left[125 x^4-50 x^2+1,2\right] \\\end{array}\right)\) | \(\begin{array}{l}\left(0,-\frac{2}{5},-\frac{2}{5},\frac{2}{5},0,0,\frac{1}{5},-\frac{2}{5}\right) \\\left(0,-\frac{2}{5},\frac{2}{5},-\frac{2}{5},\frac{1}{5},0,0,-\frac{2}{5}\right) \\\left(0,\frac{2}{5},-\frac{2}{5},-\frac{2}{5},0,\frac{1}{5},0,-\frac{2}{5}\right) \\\left(0,\frac{2}{5},\frac{2}{5},-\frac{2}{5},0,0,-\frac{1}{5},\frac{2}{5}\right) \\\left(0,\frac{2}{5},-\frac{2}{5},\frac{2}{5},-\frac{1}{5},0,0,\frac{2}{5}\right) \\\left(0,-\frac{2}{5},\frac{2}{5},\frac{2}{5},0,-\frac{1}{5},0,\frac{2}{5}\right) \\\left(0,-\frac{2}{5},-\frac{2}{5},-\frac{2}{5},\frac{1}{5},\frac{1}{5},\frac{1}{5},-\frac{1}{5}\right) \\\left(0,\frac{2}{5},\frac{2}{5},\frac{2}{5},-\frac{1}{5},-\frac{1}{5},-\frac{1}{5},\frac{1}{5}\right)\end{array}\) |
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
Download links for numeric data: