\(\text{FR}^{9,4}_{25}\)
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{4} & \mathbf{3} & \mathbf{6} & \mathbf{5} & \mathbf{8} & \mathbf{7} & \mathbf{9} \\ \mathbf{3} & \mathbf{4} & \mathbf{2} & \mathbf{1} & \mathbf{7} & \mathbf{8} & \mathbf{6} & \mathbf{5} & \mathbf{9} \\ \mathbf{4} & \mathbf{3} & \mathbf{1} & \mathbf{2} & \mathbf{8} & \mathbf{7} & \mathbf{5} & \mathbf{6} & \mathbf{9} \\ \mathbf{5} & \mathbf{6} & \mathbf{7} & \mathbf{8} & \mathbf{1}+\mathbf{6}+\mathbf{7}+\mathbf{8}+\mathbf{9} & \mathbf{2}+\mathbf{5}+\mathbf{7}+\mathbf{8}+\mathbf{9} & \mathbf{3}+\mathbf{5}+\mathbf{6}+\mathbf{8}+\mathbf{9} & \mathbf{4}+\mathbf{5}+\mathbf{6}+\mathbf{7}+\mathbf{9} & \mathbf{5}+\mathbf{6}+\mathbf{7}+\mathbf{8}+\mathbf{9} \\ \mathbf{6} & \mathbf{5} & \mathbf{8} & \mathbf{7} & \mathbf{2}+\mathbf{5}+\mathbf{7}+\mathbf{8}+\mathbf{9} & \mathbf{1}+\mathbf{6}+\mathbf{7}+\mathbf{8}+\mathbf{9} & \mathbf{4}+\mathbf{5}+\mathbf{6}+\mathbf{7}+\mathbf{9} & \mathbf{3}+\mathbf{5}+\mathbf{6}+\mathbf{8}+\mathbf{9} & \mathbf{5}+\mathbf{6}+\mathbf{7}+\mathbf{8}+\mathbf{9} \\ \mathbf{7} & \mathbf{8} & \mathbf{6} & \mathbf{5} & \mathbf{3}+\mathbf{5}+\mathbf{6}+\mathbf{8}+\mathbf{9} & \mathbf{4}+\mathbf{5}+\mathbf{6}+\mathbf{7}+\mathbf{9} & \mathbf{2}+\mathbf{5}+\mathbf{7}+\mathbf{8}+\mathbf{9} & \mathbf{1}+\mathbf{6}+\mathbf{7}+\mathbf{8}+\mathbf{9} & \mathbf{5}+\mathbf{6}+\mathbf{7}+\mathbf{8}+\mathbf{9} \\ \mathbf{8} & \mathbf{7} & \mathbf{5} & \mathbf{6} & \mathbf{4}+\mathbf{5}+\mathbf{6}+\mathbf{7}+\mathbf{9} & \mathbf{3}+\mathbf{5}+\mathbf{6}+\mathbf{8}+\mathbf{9} & \mathbf{1}+\mathbf{6}+\mathbf{7}+\mathbf{8}+\mathbf{9} & \mathbf{2}+\mathbf{5}+\mathbf{7}+\mathbf{8}+\mathbf{9} & \mathbf{5}+\mathbf{6}+\mathbf{7}+\mathbf{8}+\mathbf{9} \\ \mathbf{9} & \mathbf{9} & \mathbf{9} & \mathbf{9} & \mathbf{5}+\mathbf{6}+\mathbf{7}+\mathbf{8}+\mathbf{9} & \mathbf{5}+\mathbf{6}+\mathbf{7}+\mathbf{8}+\mathbf{9} & \mathbf{5}+\mathbf{6}+\mathbf{7}+\mathbf{8}+\mathbf{9} & \mathbf{5}+\mathbf{6}+\mathbf{7}+\mathbf{8}+\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:
\[\{(\mathbf{3} \ \mathbf{4}) (\mathbf{7} \ \mathbf{8})\}\]The following particles form non-trivial sub fusion rings
| Particles | SubRing |
|---|---|
| \(\{\mathbf{1},\mathbf{2}\}\) | \(\mathbb{Z}_2:\ \text{FR}^{2,0}_{1}\) |
| \(\{\mathbf{1},\mathbf{2},\mathbf{3},\mathbf{4}\}\) | \(\mathbb{Z}_4:\ \text{FR}^{4,2}_{1}\) |
Quantum Dimensions
| Particle | Numeric | Symbolic |
|---|---|---|
| \(\mathbf{1}\) | \(1.\) | \(1\) |
| \(\mathbf{2}\) | \(1.\) | \(1\) |
| \(\mathbf{3}\) | \(1.\) | \(1\) |
| \(\mathbf{4}\) | \(1.\) | \(1\) |
| \(\mathbf{5}\) | \(4.40268\) | \(\text{Root}\left[x^3-4 x^2-2 x+1,3\right]\) |
| \(\mathbf{6}\) | \(4.40268\) | \(\text{Root}\left[x^3-4 x^2-2 x+1,3\right]\) |
| \(\mathbf{7}\) | \(4.40268\) | \(\text{Root}\left[x^3-4 x^2-2 x+1,3\right]\) |
| \(\mathbf{8}\) | \(4.40268\) | \(\text{Root}\left[x^3-4 x^2-2 x+1,3\right]\) |
| \(\mathbf{9}\) | \(5.17554\) | \(\text{Root}\left[x^3-5 x^2-4 x+16,3\right]\) |
| \(\mathcal{D}_{FP}^2\) | \(108.321\) | \(4 \text{Root}\left[x^3-4 x^2-2 x+1,3\right]^2+\text{Root}\left[x^3-5 x^2-4 x+16,3\right]^2+4\) |
Characters
The symbolic character table is the following
\[\begin{array}{|ccccccccc|} \hline \mathbf{1} & \mathbf{2} & \mathbf{4} & \mathbf{3} & \mathbf{5} & \mathbf{7} & \mathbf{8} & \mathbf{6} & \mathbf{9} \\ \hline 1 & 1 & 1 & 1 & \text{Root}\left[x^3-4 x^2-2 x+1,3\right] & \text{Root}\left[x^3-4 x^2-2 x+1,3\right] & \text{Root}\left[x^3-4 x^2-2 x+1,3\right] & \text{Root}\left[x^3-4 x^2-2 x+1,3\right] & \text{Root}\left[x^3-5 x^2-4 x+16,3\right] \\ 1 & 1 & 1 & 1 & \text{Root}\left[x^3-4 x^2-2 x+1,2\right] & \text{Root}\left[x^3-4 x^2-2 x+1,2\right] & \text{Root}\left[x^3-4 x^2-2 x+1,2\right] & \text{Root}\left[x^3-4 x^2-2 x+1,2\right] & \text{Root}\left[x^3-5 x^2-4 x+16,1\right] \\ 1 & 1 & 1 & 1 & \text{Root}\left[x^3-4 x^2-2 x+1,1\right] & \text{Root}\left[x^3-4 x^2-2 x+1,1\right] & \text{Root}\left[x^3-4 x^2-2 x+1,1\right] & \text{Root}\left[x^3-4 x^2-2 x+1,1\right] & \text{Root}\left[x^3-5 x^2-4 x+16,2\right] \\ 1 & 1 & -1 & -1 & \frac{1}{2} \left(\sqrt{5}-1\right) & \frac{1}{2} \left(1-\sqrt{5}\right) & \frac{1}{2} \left(1-\sqrt{5}\right) & \frac{1}{2} \left(\sqrt{5}-1\right) & 0 \\ 1 & 1 & -1 & -1 & \frac{1}{2} \left(-1-\sqrt{5}\right) & \frac{1}{2} \left(1+\sqrt{5}\right) & \frac{1}{2} \left(1+\sqrt{5}\right) & \frac{1}{2} \left(-1-\sqrt{5}\right) & 0 \\ 1 & -1 & -i & i & \frac{1}{2} \left(\sqrt{5}-1\right) & \text{Root}\left[x^4+3 x^2+1,2\right] & \text{Root}\left[x^4+3 x^2+1,1\right] & \frac{1}{2} \left(1-\sqrt{5}\right) & 0 \\ 1 & -1 & i & -i & \frac{1}{2} \left(\sqrt{5}-1\right) & \text{Root}\left[x^4+3 x^2+1,1\right] & \text{Root}\left[x^4+3 x^2+1,2\right] & \frac{1}{2} \left(1-\sqrt{5}\right) & 0 \\ 1 & -1 & i & -i & \frac{1}{2} \left(-1-\sqrt{5}\right) & \text{Root}\left[x^4+3 x^2+1,4\right] & \text{Root}\left[x^4+3 x^2+1,3\right] & \frac{1}{2} \left(1+\sqrt{5}\right) & 0 \\ 1 & -1 & -i & i & \frac{1}{2} \left(-1-\sqrt{5}\right) & \text{Root}\left[x^4+3 x^2+1,3\right] & \text{Root}\left[x^4+3 x^2+1,4\right] & \frac{1}{2} \left(1+\sqrt{5}\right) & 0 \\ \hline \end{array}\]The numeric character table is the following
\[\begin{array}{|rrrrrrrrr|} \hline \mathbf{1} & \mathbf{2} & \mathbf{4} & \mathbf{3} & \mathbf{5} & \mathbf{7} & \mathbf{8} & \mathbf{6} & \mathbf{9} \\ \hline 1.000 & 1.000 & 1.000 & 1.000 & 4.403 & 4.403 & 4.403 & 4.403 & 5.176 \\ 1.000 & 1.000 & 1.000 & 1.000 & 0.3160 & 0.3160 & 0.3160 & 0.3160 & -1.848 \\ 1.000 & 1.000 & 1.000 & 1.000 & -0.7187 & -0.7187 & -0.7187 & -0.7187 & 1.673 \\ 1.000 & 1.000 & -1.000 & -1.000 & 0.6180 & -0.6180 & -0.6180 & 0.6180 & 0 \\ 1.000 & 1.000 & -1.000 & -1.000 & -1.618 & 1.618 & 1.618 & -1.618 & 0 \\ 1.000 & -1.000 & -1.000 i & 1.000 i & 0.6180 & \text{0$\grave{ }\grave{ }$4.35950263808197}+0.6180 i & \text{0$\grave{ }\grave{ }$4.35950263808197}-0.6180 i & -0.6180 & 0 \\ 1.000 & -1.000 & 1.000 i & -1.000 i & 0.6180 & \text{0$\grave{ }\grave{ }$4.35950263808197}-0.6180 i & \text{0$\grave{ }\grave{ }$4.35950263808197}+0.6180 i & -0.6180 & 0 \\ 1.000 & -1.000 & 1.000 i & -1.000 i & -1.618 & \text{0$\grave{ }\grave{ }$3.9415273575820122}+1.618 i & \text{0$\grave{ }\grave{ }$3.9415273575820122}-1.618 i & 1.618 & 0 \\ 1.000 & -1.000 & -1.000 i & 1.000 i & -1.618 & \text{0$\grave{ }\grave{ }$3.9415273575820122}-1.618 i & \text{0$\grave{ }\grave{ }$3.9415273575820122}+1.618 i & 1.618 & 0 \\ \hline \end{array}\]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 extended cyclotomic criterion.
Data
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