\(\text{FR}^{9,0}_{46}\)
Fusion Rules
\[\begin{array}{|lllllllll|} \hline \mathbf{1} & \mathbf{2} & \mathbf{3} & \mathbf{4} & \mathbf{5} & \mathbf{6} & \mathbf{7} & \mathbf{8} & \mathbf{9} \\ \mathbf{2} & \mathbf{1}+\mathbf{2}+\mathbf{4}+\mathbf{5}+\mathbf{6}+\mathbf{7}+\mathbf{8} & \mathbf{4}+\mathbf{5}+\mathbf{6}+\mathbf{7}+\mathbf{8}+\mathbf{9} & \mathbf{2}+\mathbf{3}+\mathbf{4}+\mathbf{5}+\mathbf{6}+\mathbf{9} & \mathbf{2}+\mathbf{3}+\mathbf{4}+\mathbf{5}+\mathbf{7}+\mathbf{9} & \mathbf{2}+\mathbf{3}+\mathbf{4}+\mathbf{6}+\mathbf{8}+\mathbf{9} & \mathbf{2}+\mathbf{3}+\mathbf{5}+\mathbf{7}+\mathbf{8}+\mathbf{9} & \mathbf{2}+\mathbf{3}+\mathbf{6}+\mathbf{7}+\mathbf{8}+\mathbf{9} & \mathbf{3}+\mathbf{4}+\mathbf{5}+\mathbf{6}+\mathbf{7}+\mathbf{8}+\mathbf{9} \\ \mathbf{3} & \mathbf{4}+\mathbf{5}+\mathbf{6}+\mathbf{7}+\mathbf{8}+\mathbf{9} & \mathbf{1}+\mathbf{3}+\mathbf{4}+\mathbf{5}+\mathbf{6}+\mathbf{7}+\mathbf{8} & \mathbf{2}+\mathbf{3}+\mathbf{4}+\mathbf{7}+\mathbf{8}+\mathbf{9} & \mathbf{2}+\mathbf{3}+\mathbf{5}+\mathbf{6}+\mathbf{8}+\mathbf{9} & \mathbf{2}+\mathbf{3}+\mathbf{5}+\mathbf{6}+\mathbf{7}+\mathbf{9} & \mathbf{2}+\mathbf{3}+\mathbf{4}+\mathbf{6}+\mathbf{7}+\mathbf{9} & \mathbf{2}+\mathbf{3}+\mathbf{4}+\mathbf{5}+\mathbf{8}+\mathbf{9} & \mathbf{2}+\mathbf{4}+\mathbf{5}+\mathbf{6}+\mathbf{7}+\mathbf{8}+\mathbf{9} \\ \mathbf{4} & \mathbf{2}+\mathbf{3}+\mathbf{4}+\mathbf{5}+\mathbf{6}+\mathbf{9} & \mathbf{2}+\mathbf{3}+\mathbf{4}+\mathbf{7}+\mathbf{8}+\mathbf{9} & \mathbf{1}+\mathbf{2}+\mathbf{3}+\mathbf{5}+\mathbf{6}+\mathbf{7}+\mathbf{8} & \mathbf{2}+\mathbf{4}+\mathbf{5}+\mathbf{7}+\mathbf{8}+\mathbf{9} & \mathbf{2}+\mathbf{4}+\mathbf{6}+\mathbf{7}+\mathbf{8}+\mathbf{9} & \mathbf{3}+\mathbf{4}+\mathbf{5}+\mathbf{6}+\mathbf{7}+\mathbf{9} & \mathbf{3}+\mathbf{4}+\mathbf{5}+\mathbf{6}+\mathbf{8}+\mathbf{9} & \mathbf{2}+\mathbf{3}+\mathbf{5}+\mathbf{6}+\mathbf{7}+\mathbf{8}+\mathbf{9} \\ \mathbf{5} & \mathbf{2}+\mathbf{3}+\mathbf{4}+\mathbf{5}+\mathbf{7}+\mathbf{9} & \mathbf{2}+\mathbf{3}+\mathbf{5}+\mathbf{6}+\mathbf{8}+\mathbf{9} & \mathbf{2}+\mathbf{4}+\mathbf{5}+\mathbf{7}+\mathbf{8}+\mathbf{9} & \mathbf{1}+\mathbf{2}+\mathbf{3}+\mathbf{4}+\mathbf{5}+\mathbf{6}+\mathbf{7} & \mathbf{3}+\mathbf{5}+\mathbf{6}+\mathbf{7}+\mathbf{8}+\mathbf{9} & \mathbf{2}+\mathbf{4}+\mathbf{5}+\mathbf{6}+\mathbf{8}+\mathbf{9} & \mathbf{3}+\mathbf{4}+\mathbf{6}+\mathbf{7}+\mathbf{8}+\mathbf{9} & \mathbf{2}+\mathbf{3}+\mathbf{4}+\mathbf{6}+\mathbf{7}+\mathbf{8}+\mathbf{9} \\ \mathbf{6} & \mathbf{2}+\mathbf{3}+\mathbf{4}+\mathbf{6}+\mathbf{8}+\mathbf{9} & \mathbf{2}+\mathbf{3}+\mathbf{5}+\mathbf{6}+\mathbf{7}+\mathbf{9} & \mathbf{2}+\mathbf{4}+\mathbf{6}+\mathbf{7}+\mathbf{8}+\mathbf{9} & \mathbf{3}+\mathbf{5}+\mathbf{6}+\mathbf{7}+\mathbf{8}+\mathbf{9} & \mathbf{1}+\mathbf{2}+\mathbf{3}+\mathbf{4}+\mathbf{5}+\mathbf{6}+\mathbf{8} & \mathbf{3}+\mathbf{4}+\mathbf{5}+\mathbf{7}+\mathbf{8}+\mathbf{9} & \mathbf{2}+\mathbf{4}+\mathbf{5}+\mathbf{6}+\mathbf{7}+\mathbf{9} & \mathbf{2}+\mathbf{3}+\mathbf{4}+\mathbf{5}+\mathbf{7}+\mathbf{8}+\mathbf{9} \\ \mathbf{7} & \mathbf{2}+\mathbf{3}+\mathbf{5}+\mathbf{7}+\mathbf{8}+\mathbf{9} & \mathbf{2}+\mathbf{3}+\mathbf{4}+\mathbf{6}+\mathbf{7}+\mathbf{9} & \mathbf{3}+\mathbf{4}+\mathbf{5}+\mathbf{6}+\mathbf{7}+\mathbf{9} & \mathbf{2}+\mathbf{4}+\mathbf{5}+\mathbf{6}+\mathbf{8}+\mathbf{9} & \mathbf{3}+\mathbf{4}+\mathbf{5}+\mathbf{7}+\mathbf{8}+\mathbf{9} & \mathbf{1}+\mathbf{2}+\mathbf{3}+\mathbf{4}+\mathbf{6}+\mathbf{7}+\mathbf{8} & \mathbf{2}+\mathbf{5}+\mathbf{6}+\mathbf{7}+\mathbf{8}+\mathbf{9} & \mathbf{2}+\mathbf{3}+\mathbf{4}+\mathbf{5}+\mathbf{6}+\mathbf{8}+\mathbf{9} \\ \mathbf{8} & \mathbf{2}+\mathbf{3}+\mathbf{6}+\mathbf{7}+\mathbf{8}+\mathbf{9} & \mathbf{2}+\mathbf{3}+\mathbf{4}+\mathbf{5}+\mathbf{8}+\mathbf{9} & \mathbf{3}+\mathbf{4}+\mathbf{5}+\mathbf{6}+\mathbf{8}+\mathbf{9} & \mathbf{3}+\mathbf{4}+\mathbf{6}+\mathbf{7}+\mathbf{8}+\mathbf{9} & \mathbf{2}+\mathbf{4}+\mathbf{5}+\mathbf{6}+\mathbf{7}+\mathbf{9} & \mathbf{2}+\mathbf{5}+\mathbf{6}+\mathbf{7}+\mathbf{8}+\mathbf{9} & \mathbf{1}+\mathbf{2}+\mathbf{3}+\mathbf{4}+\mathbf{5}+\mathbf{7}+\mathbf{8} & \mathbf{2}+\mathbf{3}+\mathbf{4}+\mathbf{5}+\mathbf{6}+\mathbf{7}+\mathbf{9} \\ \mathbf{9} & \mathbf{3}+\mathbf{4}+\mathbf{5}+\mathbf{6}+\mathbf{7}+\mathbf{8}+\mathbf{9} & \mathbf{2}+\mathbf{4}+\mathbf{5}+\mathbf{6}+\mathbf{7}+\mathbf{8}+\mathbf{9} & \mathbf{2}+\mathbf{3}+\mathbf{5}+\mathbf{6}+\mathbf{7}+\mathbf{8}+\mathbf{9} & \mathbf{2}+\mathbf{3}+\mathbf{4}+\mathbf{6}+\mathbf{7}+\mathbf{8}+\mathbf{9} & \mathbf{2}+\mathbf{3}+\mathbf{4}+\mathbf{5}+\mathbf{7}+\mathbf{8}+\mathbf{9} & \mathbf{2}+\mathbf{3}+\mathbf{4}+\mathbf{5}+\mathbf{6}+\mathbf{8}+\mathbf{9} & \mathbf{2}+\mathbf{3}+\mathbf{4}+\mathbf{5}+\mathbf{6}+\mathbf{7}+\mathbf{9} & \mathbf{1}+\mathbf{2}+\mathbf{3}+\mathbf{4}+\mathbf{5}+\mathbf{6}+\mathbf{7}+\mathbf{8}+\mathbf{9} \\ \hline \end{array}\]The fusion rules are invariant under the group generated by the following permutations:
\[\{(\mathbf{2} \ \mathbf{3}) (\mathbf{5} \ \mathbf{7} \ \mathbf{6} \ \mathbf{8}), (\mathbf{2} \ \mathbf{3}) (\mathbf{5} \ \mathbf{8} \ \mathbf{6} \ \mathbf{7})\}\]Quantum Dimensions
| Particle | Numeric | Symbolic |
|---|---|---|
| \(\mathbf{1}\) | \(1.\) | \(1\) |
| \(\mathbf{2}\) | \(6.16228\) | \(3+\sqrt{10}\) |
| \(\mathbf{3}\) | \(6.16228\) | \(3+\sqrt{10}\) |
| \(\mathbf{4}\) | \(6.16228\) | \(3+\sqrt{10}\) |
| \(\mathbf{5}\) | \(6.16228\) | \(3+\sqrt{10}\) |
| \(\mathbf{6}\) | \(6.16228\) | \(3+\sqrt{10}\) |
| \(\mathbf{7}\) | \(6.16228\) | \(3+\sqrt{10}\) |
| \(\mathbf{8}\) | \(6.16228\) | \(3+\sqrt{10}\) |
| \(\mathbf{9}\) | \(7.16228\) | \(4+\sqrt{10}\) |
| \(\mathcal{D}_{FP}^2\) | \(318.114\) | \(1+7 \left(3+\sqrt{10}\right)^2+\left(4+\sqrt{10}\right)^2\) |
Characters
The symbolic character table is the following
\[\begin{array}{|ccccccccc|} \hline \mathbf{1} & \mathbf{2} & \mathbf{3} & \mathbf{4} & \mathbf{5} & \mathbf{6} & \mathbf{7} & \mathbf{8} & \mathbf{9} \\ \hline 1 & 3+\sqrt{10} & 3+\sqrt{10} & 3+\sqrt{10} & 3+\sqrt{10} & 3+\sqrt{10} & 3+\sqrt{10} & 3+\sqrt{10} & 4+\sqrt{10} \\ 1 & \frac{1}{2} \left(1-\sqrt{5}\right) & \frac{1}{2} \left(1+\sqrt{5}\right) & 1 & \text{Root}\left[x^4+x^3-4 x^2-4 x+1,3\right] & \text{Root}\left[x^4+x^3-4 x^2-4 x+1,1\right] & \text{Root}\left[x^4+x^3-4 x^2-4 x+1,2\right] & \text{Root}\left[x^4+x^3-4 x^2-4 x+1,4\right] & -1 \\ 1 & \frac{1}{2} \left(1+\sqrt{5}\right) & \frac{1}{2} \left(1-\sqrt{5}\right) & 1 & \text{Root}\left[x^4+x^3-4 x^2-4 x+1,4\right] & \text{Root}\left[x^4+x^3-4 x^2-4 x+1,2\right] & \text{Root}\left[x^4+x^3-4 x^2-4 x+1,3\right] & \text{Root}\left[x^4+x^3-4 x^2-4 x+1,1\right] & -1 \\ 1 & \frac{1}{2} \left(1-\sqrt{5}\right) & \frac{1}{2} \left(1+\sqrt{5}\right) & 1 & \text{Root}\left[x^4+x^3-4 x^2-4 x+1,1\right] & \text{Root}\left[x^4+x^3-4 x^2-4 x+1,3\right] & \text{Root}\left[x^4+x^3-4 x^2-4 x+1,4\right] & \text{Root}\left[x^4+x^3-4 x^2-4 x+1,2\right] & -1 \\ 1 & \frac{1}{2} \left(1+\sqrt{5}\right) & \frac{1}{2} \left(1-\sqrt{5}\right) & 1 & \text{Root}\left[x^4+x^3-4 x^2-4 x+1,2\right] & \text{Root}\left[x^4+x^3-4 x^2-4 x+1,4\right] & \text{Root}\left[x^4+x^3-4 x^2-4 x+1,1\right] & \text{Root}\left[x^4+x^3-4 x^2-4 x+1,3\right] & -1 \\ 1 & \frac{1}{2} \left(1+\sqrt{5}\right) & \frac{1}{2} \left(1-\sqrt{5}\right) & -2 & \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(1+\sqrt{5}\right) & -1 \\ 1 & -2 & -2 & 1 & 1 & 1 & 1 & 1 & -1 \\ 1 & \frac{1}{2} \left(1-\sqrt{5}\right) & \frac{1}{2} \left(1+\sqrt{5}\right) & -2 & \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(1-\sqrt{5}\right) & -1 \\ 1 & 3-\sqrt{10} & 3-\sqrt{10} & 3-\sqrt{10} & 3-\sqrt{10} & 3-\sqrt{10} & 3-\sqrt{10} & 3-\sqrt{10} & 4-\sqrt{10} \\ \hline \end{array}\]The numeric character table is the following
\[\begin{array}{|rrrrrrrrr|} \hline \mathbf{1} & \mathbf{2} & \mathbf{3} & \mathbf{4} & \mathbf{5} & \mathbf{6} & \mathbf{7} & \mathbf{8} & \mathbf{9} \\ \hline 1.000 & 6.162 & 6.162 & 6.162 & 6.162 & 6.162 & 6.162 & 6.162 & 7.162 \\ 1.000 & -0.6180 & 1.618 & 1.000 & 0.2091 & -1.827 & -1.338 & 1.956 & -1.000 \\ 1.000 & 1.618 & -0.6180 & 1.000 & 1.956 & -1.338 & 0.2091 & -1.827 & -1.000 \\ 1.000 & -0.6180 & 1.618 & 1.000 & -1.827 & 0.2091 & 1.956 & -1.338 & -1.000 \\ 1.000 & 1.618 & -0.6180 & 1.000 & -1.338 & 1.956 & -1.827 & 0.2091 & -1.000 \\ 1.000 & 1.618 & -0.6180 & -2.000 & -0.6180 & -0.6180 & 1.618 & 1.618 & -1.000 \\ 1.000 & -2.000 & -2.000 & 1.000 & 1.000 & 1.000 & 1.000 & 1.000 & -1.000 \\ 1.000 & -0.6180 & 1.618 & -2.000 & 1.618 & 1.618 & -0.6180 & -0.6180 & -1.000 \\ 1.000 & -0.1623 & -0.1623 & -0.1623 & -0.1623 & -0.1623 & -0.1623 & -0.1623 & 0.8377 \\ \hline \end{array}\]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
This fusion ring has no categorifications because of the $d$-number criterion.
Data
Download links for numeric data: