\(\text{FR}^{9,0}_{36}\)
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{7} & \mathbf{9} & \mathbf{8} \\ \mathbf{3} & \mathbf{4} & \mathbf{1}+\mathbf{4}+\mathbf{7} & \mathbf{2}+\mathbf{3}+\mathbf{7} & \mathbf{6}+\mathbf{9} & \mathbf{5}+\mathbf{8} & \mathbf{3}+\mathbf{4}+\mathbf{7} & \mathbf{6}+\mathbf{8}+\mathbf{9} & \mathbf{5}+\mathbf{8}+\mathbf{9} \\ \mathbf{4} & \mathbf{3} & \mathbf{2}+\mathbf{3}+\mathbf{7} & \mathbf{1}+\mathbf{4}+\mathbf{7} & \mathbf{5}+\mathbf{8} & \mathbf{6}+\mathbf{9} & \mathbf{3}+\mathbf{4}+\mathbf{7} & \mathbf{5}+\mathbf{8}+\mathbf{9} & \mathbf{6}+\mathbf{8}+\mathbf{9} \\ \mathbf{5} & \mathbf{6} & \mathbf{6}+\mathbf{9} & \mathbf{5}+\mathbf{8} & \mathbf{1}+\mathbf{4}+\mathbf{5}+\mathbf{6} & \mathbf{2}+\mathbf{3}+\mathbf{5}+\mathbf{6} & \mathbf{8}+\mathbf{9} & \mathbf{4}+\mathbf{7}+\mathbf{8}+\mathbf{9} & \mathbf{3}+\mathbf{7}+\mathbf{8}+\mathbf{9} \\ \mathbf{6} & \mathbf{5} & \mathbf{5}+\mathbf{8} & \mathbf{6}+\mathbf{9} & \mathbf{2}+\mathbf{3}+\mathbf{5}+\mathbf{6} & \mathbf{1}+\mathbf{4}+\mathbf{5}+\mathbf{6} & \mathbf{8}+\mathbf{9} & \mathbf{3}+\mathbf{7}+\mathbf{8}+\mathbf{9} & \mathbf{4}+\mathbf{7}+\mathbf{8}+\mathbf{9} \\ \mathbf{7} & \mathbf{7} & \mathbf{3}+\mathbf{4}+\mathbf{7} & \mathbf{3}+\mathbf{4}+\mathbf{7} & \mathbf{8}+\mathbf{9} & \mathbf{8}+\mathbf{9} & \mathbf{1}+\mathbf{2}+\mathbf{3}+\mathbf{4}+\mathbf{7} & \mathbf{5}+\mathbf{6}+\mathbf{8}+\mathbf{9} & \mathbf{5}+\mathbf{6}+\mathbf{8}+\mathbf{9} \\ \mathbf{8} & \mathbf{9} & \mathbf{6}+\mathbf{8}+\mathbf{9} & \mathbf{5}+\mathbf{8}+\mathbf{9} & \mathbf{4}+\mathbf{7}+\mathbf{8}+\mathbf{9} & \mathbf{3}+\mathbf{7}+\mathbf{8}+\mathbf{9} & \mathbf{5}+\mathbf{6}+\mathbf{8}+\mathbf{9} & \mathbf{1}+\mathbf{3}+\mathbf{4}+\mathbf{5}+\mathbf{6}+\mathbf{7}+\mathbf{8}+\mathbf{9} & \mathbf{2}+\mathbf{3}+\mathbf{4}+\mathbf{5}+\mathbf{6}+\mathbf{7}+\mathbf{8}+\mathbf{9} \\ \mathbf{9} & \mathbf{8} & \mathbf{5}+\mathbf{8}+\mathbf{9} & \mathbf{6}+\mathbf{8}+\mathbf{9} & \mathbf{3}+\mathbf{7}+\mathbf{8}+\mathbf{9} & \mathbf{4}+\mathbf{7}+\mathbf{8}+\mathbf{9} & \mathbf{5}+\mathbf{6}+\mathbf{8}+\mathbf{9} & \mathbf{2}+\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{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 |
|---|---|
| \(\{\mathbf{1},\mathbf{2}\}\) | \(\mathbb{Z}_2:\ \text{FR}^{2,0}_{1}\) |
| \(\{\mathbf{1},\mathbf{2},\mathbf{3},\mathbf{4},\mathbf{7}\}\) | \(\text{PSU(2})_8:\ \text{FR}^{5,0}_{7}\) |
Quantum Dimensions
| Particle | Numeric | Symbolic |
|---|---|---|
| \(\mathbf{1}\) | \(1.\) | \(1\) |
| \(\mathbf{2}\) | \(1.\) | \(1\) |
| \(\mathbf{3}\) | \(2.61803\) | \(\frac{1}{2} \left(3+\sqrt{5}\right)\) |
| \(\mathbf{4}\) | \(2.61803\) | \(\frac{1}{2} \left(3+\sqrt{5}\right)\) |
| \(\mathbf{5}\) | \(3.14896\) | \(\frac{1}{2} \left(2+\sqrt{2 \left(7+\sqrt{5}\right)}\right)\) |
| \(\mathbf{6}\) | \(3.14896\) | \(\frac{1}{2} \left(2+\sqrt{2 \left(7+\sqrt{5}\right)}\right)\) |
| \(\mathbf{7}\) | \(3.23607\) | \(1+\sqrt{5}\) |
| \(\mathbf{8}\) | \(5.09513\) | \(\text{Root}\left[x^4-2 x^3-14 x^2-10 x+5,4\right]\) |
| \(\mathbf{9}\) | \(5.09513\) | \(\text{Root}\left[x^4-2 x^3-14 x^2-10 x+5,4\right]\) |
| \(\mathcal{D}_{FP}^2\) | \(97.9329\) | \(2 \text{Root}\left[x^4-2 x^3-14 x^2-10 x+5,4\right]^2+2+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2+\frac{1}{2} \left(2+\sqrt{2 \left(7+\sqrt{5}\right)}\right)^2\) |
Characters
The symbolic character table is the following
\[\begin{array}{|ccccccccc|} \hline \mathbf{1} & \mathbf{2} & \mathbf{4} & \mathbf{3} & \mathbf{5} & \mathbf{6} & \mathbf{7} & \mathbf{9} & \mathbf{8} \\ \hline 1 & 1 & \frac{1}{2} \left(3+\sqrt{5}\right) & \frac{1}{2} \left(3+\sqrt{5}\right) & \text{Root}\left[x^4-4 x^3-x^2+10 x+5,4\right] & \text{Root}\left[x^4-4 x^3-x^2+10 x+5,4\right] & 1+\sqrt{5} & \text{Root}\left[x^4-2 x^3-14 x^2-10 x+5,4\right] & \text{Root}\left[x^4-2 x^3-14 x^2-10 x+5,4\right] \\ 1 & 1 & \frac{1}{2} \left(3+\sqrt{5}\right) & \frac{1}{2} \left(3+\sqrt{5}\right) & \text{Root}\left[x^4-4 x^3-x^2+10 x+5,1\right] & \text{Root}\left[x^4-4 x^3-x^2+10 x+5,1\right] & 1+\sqrt{5} & \text{Root}\left[x^4-2 x^3-14 x^2-10 x+5,1\right] & \text{Root}\left[x^4-2 x^3-14 x^2-10 x+5,1\right] \\ 1 & 1 & \frac{1}{2} \left(3-\sqrt{5}\right) & \frac{1}{2} \left(3-\sqrt{5}\right) & \text{Root}\left[x^4-4 x^3-x^2+10 x+5,3\right] & \text{Root}\left[x^4-4 x^3-x^2+10 x+5,3\right] & 1-\sqrt{5} & \text{Root}\left[x^4-2 x^3-14 x^2-10 x+5,2\right] & \text{Root}\left[x^4-2 x^3-14 x^2-10 x+5,2\right] \\ 1 & 1 & \frac{1}{2} \left(3-\sqrt{5}\right) & \frac{1}{2} \left(3-\sqrt{5}\right) & \text{Root}\left[x^4-4 x^3-x^2+10 x+5,2\right] & \text{Root}\left[x^4-4 x^3-x^2+10 x+5,2\right] & 1-\sqrt{5} & \text{Root}\left[x^4-2 x^3-14 x^2-10 x+5,3\right] & \text{Root}\left[x^4-2 x^3-14 x^2-10 x+5,3\right] \\ 1 & 1 & -1 & -1 & 0 & 0 & 1 & 0 & 0 \\ 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 \\ 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 \\ 1 & -1 & \frac{1}{2} \left(1-\sqrt{5}\right) & \frac{1}{2} \left(\sqrt{5}-1\right) & \frac{1}{2} \left(\sqrt{5}-1\right) & \frac{1}{2} \left(1-\sqrt{5}\right) & 0 & 1 & -1 \\ 1 & -1 & \frac{1}{2} \left(1-\sqrt{5}\right) & \frac{1}{2} \left(\sqrt{5}-1\right) & \frac{1}{2} \left(1-\sqrt{5}\right) & \frac{1}{2} \left(\sqrt{5}-1\right) & 0 & -1 & 1 \\ \hline \end{array}\]The numeric character table is the following
\[\begin{array}{|rrrrrrrrr|} \hline \mathbf{1} & \mathbf{2} & \mathbf{4} & \mathbf{3} & \mathbf{5} & \mathbf{6} & \mathbf{7} & \mathbf{9} & \mathbf{8} \\ \hline 1.000 & 1.000 & 2.618 & 2.618 & 3.149 & 3.149 & 3.236 & 5.095 & 5.095 \\ 1.000 & 1.000 & 2.618 & 2.618 & -1.149 & -1.149 & 3.236 & -1.859 & -1.859 \\ 1.000 & 1.000 & 0.3820 & 0.3820 & 2.543 & 2.543 & -1.236 & -1.572 & -1.572 \\ 1.000 & 1.000 & 0.3820 & 0.3820 & -0.5434 & -0.5434 & -1.236 & 0.3358 & 0.3358 \\ 1.000 & 1.000 & -1.000 & -1.000 & 0 & 0 & 1.000 & 0 & 0 \\ 1.000 & -1.000 & 1.618 & -1.618 & 1.618 & -1.618 & 0 & -1.000 & 1.000 \\ 1.000 & -1.000 & 1.618 & -1.618 & -1.618 & 1.618 & 0 & 1.000 & -1.000 \\ 1.000 & -1.000 & -0.6180 & 0.6180 & 0.6180 & -0.6180 & 0 & 1.000 & -1.000 \\ 1.000 & -1.000 & -0.6180 & 0.6180 & -0.6180 & 0.6180 & 0 & -1.000 & 1.000 \\ \hline \end{array}\]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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