\(\text{FR}^{9,0}_{39}\)
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{3} & \mathbf{4} & \mathbf{5} & \mathbf{6} & \mathbf{7} & \mathbf{8} & \mathbf{9} \\ \mathbf{3} & \mathbf{3} & \mathbf{1}+\mathbf{2}+\mathbf{3} & \mathbf{5}+\mathbf{6} & \mathbf{4}+\mathbf{6} & \mathbf{4}+\mathbf{5} & \mathbf{8}+\mathbf{9} & \mathbf{7}+\mathbf{9} & \mathbf{7}+\mathbf{8} \\ \mathbf{4} & \mathbf{4} & \mathbf{5}+\mathbf{6} & \mathbf{1}+\mathbf{2}+\mathbf{8}+\mathbf{9} & \mathbf{3}+\mathbf{7}+\mathbf{9} & \mathbf{3}+\mathbf{7}+\mathbf{8} & \mathbf{5}+\mathbf{6}+\mathbf{8}+\mathbf{9} & \mathbf{4}+\mathbf{6}+\mathbf{7}+\mathbf{8} & \mathbf{4}+\mathbf{5}+\mathbf{7}+\mathbf{9} \\ \mathbf{5} & \mathbf{5} & \mathbf{4}+\mathbf{6} & \mathbf{3}+\mathbf{7}+\mathbf{9} & \mathbf{1}+\mathbf{2}+\mathbf{7}+\mathbf{8} & \mathbf{3}+\mathbf{8}+\mathbf{9} & \mathbf{4}+\mathbf{5}+\mathbf{7}+\mathbf{8} & \mathbf{5}+\mathbf{6}+\mathbf{7}+\mathbf{9} & \mathbf{4}+\mathbf{6}+\mathbf{8}+\mathbf{9} \\ \mathbf{6} & \mathbf{6} & \mathbf{4}+\mathbf{5} & \mathbf{3}+\mathbf{7}+\mathbf{8} & \mathbf{3}+\mathbf{8}+\mathbf{9} & \mathbf{1}+\mathbf{2}+\mathbf{7}+\mathbf{9} & \mathbf{4}+\mathbf{6}+\mathbf{7}+\mathbf{9} & \mathbf{4}+\mathbf{5}+\mathbf{8}+\mathbf{9} & \mathbf{5}+\mathbf{6}+\mathbf{7}+\mathbf{8} \\ \mathbf{7} & \mathbf{7} & \mathbf{8}+\mathbf{9} & \mathbf{5}+\mathbf{6}+\mathbf{8}+\mathbf{9} & \mathbf{4}+\mathbf{5}+\mathbf{7}+\mathbf{8} & \mathbf{4}+\mathbf{6}+\mathbf{7}+\mathbf{9} & \mathbf{1}+\mathbf{2}+\mathbf{5}+\mathbf{6}+\mathbf{7}+\mathbf{8}+\mathbf{9} & \mathbf{3}+\mathbf{4}+\mathbf{5}+\mathbf{7}+\mathbf{8}+\mathbf{9} & \mathbf{3}+\mathbf{4}+\mathbf{6}+\mathbf{7}+\mathbf{8}+\mathbf{9} \\ \mathbf{8} & \mathbf{8} & \mathbf{7}+\mathbf{9} & \mathbf{4}+\mathbf{6}+\mathbf{7}+\mathbf{8} & \mathbf{5}+\mathbf{6}+\mathbf{7}+\mathbf{9} & \mathbf{4}+\mathbf{5}+\mathbf{8}+\mathbf{9} & \mathbf{3}+\mathbf{4}+\mathbf{5}+\mathbf{7}+\mathbf{8}+\mathbf{9} & \mathbf{1}+\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{9} & \mathbf{9} & \mathbf{7}+\mathbf{8} & \mathbf{4}+\mathbf{5}+\mathbf{7}+\mathbf{9} & \mathbf{4}+\mathbf{6}+\mathbf{8}+\mathbf{9} & \mathbf{5}+\mathbf{6}+\mathbf{7}+\mathbf{8} & \mathbf{3}+\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{4}+\mathbf{5}+\mathbf{7}+\mathbf{8}+\mathbf{9} \\ \hline \end{array}\]The fusion rules are invariant under the group generated by the following permutations:
\[\{(\mathbf{5} \ \mathbf{6}) (\mathbf{8} \ \mathbf{9})\}\]The following particles form non-trivial sub fusion rings
| Particles | SubRing |
|---|---|
| \(\{\mathbf{1},\mathbf{2}\}\) | \(\mathbb{Z}_2:\ \text{FR}^{2,0}_{1}\) |
| \(\{\mathbf{1},\mathbf{2},\mathbf{3}\}\) | \(\left.\text{Rep(}D_3\right):\ \text{FR}^{3,0}_{2}\) |
Quantum Dimensions
| Particle | Numeric | Symbolic |
|---|---|---|
| \(\mathbf{1}\) | \(1.\) | \(1\) |
| \(\mathbf{2}\) | \(1.\) | \(1\) |
| \(\mathbf{3}\) | \(2.\) | \(2\) |
| \(\mathbf{4}\) | \(3.41421\) | \(2+\sqrt{2}\) |
| \(\mathbf{5}\) | \(3.41421\) | \(2+\sqrt{2}\) |
| \(\mathbf{6}\) | \(3.41421\) | \(2+\sqrt{2}\) |
| \(\mathbf{7}\) | \(4.82843\) | \(2 \left(1+\sqrt{2}\right)\) |
| \(\mathbf{8}\) | \(4.82843\) | \(2 \left(1+\sqrt{2}\right)\) |
| \(\mathbf{9}\) | \(4.82843\) | \(2 \left(1+\sqrt{2}\right)\) |
| \(\mathcal{D}_{FP}^2\) | \(110.912\) | \(6+12 \left(1+\sqrt{2}\right)^2+3 \left(2+\sqrt{2}\right)^2\) |
Characters
The symbolic character table is the following
\[\begin{array}{|ccccccccc|} \hline \mathbf{1} & \mathbf{2} & \mathbf{3} & \mathbf{4} & \mathbf{5} & \mathbf{6} & \mathbf{7} & \mathbf{8} & \mathbf{9} \\ \hline 1 & 1 & 2 & 2+\sqrt{2} & 2+\sqrt{2} & 2+\sqrt{2} & 2+2 \sqrt{2} & 2+2 \sqrt{2} & 2+2 \sqrt{2} \\ 1 & 1 & 2 & 2-\sqrt{2} & 2-\sqrt{2} & 2-\sqrt{2} & 2-2 \sqrt{2} & 2-2 \sqrt{2} & 2-2 \sqrt{2} \\ 1 & 1 & -1 & \frac{1}{2} \left(1+\sqrt{5}\right) & \text{Root}\left[x^4+x^3-4 x^2-4 x+1,3\right] & \text{Root}\left[x^4+x^3-4 x^2-4 x+1,1\right] & \frac{1}{2} \left(1-\sqrt{5}\right) & \text{Root}\left[x^4+x^3-4 x^2-4 x+1,2\right] & \text{Root}\left[x^4+x^3-4 x^2-4 x+1,4\right] \\ 1 & 1 & 2 & -2 & -2 & -2 & 1 & 1 & 1 \\ 1 & 1 & -1 & -2 & 1 & 1 & -2 & 1 & 1 \\ 1 & 1 & -1 & \frac{1}{2} \left(1+\sqrt{5}\right) & \text{Root}\left[x^4+x^3-4 x^2-4 x+1,1\right] & \text{Root}\left[x^4+x^3-4 x^2-4 x+1,3\right] & \frac{1}{2} \left(1-\sqrt{5}\right) & \text{Root}\left[x^4+x^3-4 x^2-4 x+1,4\right] & \text{Root}\left[x^4+x^3-4 x^2-4 x+1,2\right] \\ 1 & 1 & -1 & \frac{1}{2} \left(1-\sqrt{5}\right) & \text{Root}\left[x^4+x^3-4 x^2-4 x+1,2\right] & \text{Root}\left[x^4+x^3-4 x^2-4 x+1,4\right] & \frac{1}{2} \left(1+\sqrt{5}\right) & \text{Root}\left[x^4+x^3-4 x^2-4 x+1,1\right] & \text{Root}\left[x^4+x^3-4 x^2-4 x+1,3\right] \\ 1 & 1 & -1 & \frac{1}{2} \left(1-\sqrt{5}\right) & \text{Root}\left[x^4+x^3-4 x^2-4 x+1,4\right] & \text{Root}\left[x^4+x^3-4 x^2-4 x+1,2\right] & \frac{1}{2} \left(1+\sqrt{5}\right) & \text{Root}\left[x^4+x^3-4 x^2-4 x+1,3\right] & \text{Root}\left[x^4+x^3-4 x^2-4 x+1,1\right] \\ 1 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ \hline \end{array}\]The numeric character table is the following
\[\begin{array}{|rrrrrrrrr|} \hline \mathbf{1} & \mathbf{2} & \mathbf{3} & \mathbf{4} & \mathbf{5} & \mathbf{6} & \mathbf{7} & \mathbf{8} & \mathbf{9} \\ \hline 1.000 & 1.000 & 2.000 & 3.414 & 3.414 & 3.414 & 4.828 & 4.828 & 4.828 \\ 1.000 & 1.000 & 2.000 & 0.5858 & 0.5858 & 0.5858 & -0.8284 & -0.8284 & -0.8284 \\ 1.000 & 1.000 & -1.000 & 1.618 & 0.2091 & -1.827 & -0.6180 & -1.338 & 1.956 \\ 1.000 & 1.000 & 2.000 & -2.000 & -2.000 & -2.000 & 1.000 & 1.000 & 1.000 \\ 1.000 & 1.000 & -1.000 & -2.000 & 1.000 & 1.000 & -2.000 & 1.000 & 1.000 \\ 1.000 & 1.000 & -1.000 & 1.618 & -1.827 & 0.2091 & -0.6180 & 1.956 & -1.338 \\ 1.000 & 1.000 & -1.000 & -0.6180 & -1.338 & 1.956 & 1.618 & -1.827 & 0.2091 \\ 1.000 & 1.000 & -1.000 & -0.6180 & 1.956 & -1.338 & 1.618 & 0.2091 & -1.827 \\ 1.000 & -1.000 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ \hline \end{array}\]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: