\(\text{FR}^{9,0}_{21}\)
Fusion Rules
\[\begin{array}{|lllllllll|} \hline \mathbf{1} & \mathbf{2} & \mathbf{3} & \mathbf{4} & \mathbf{5} & \mathbf{6} & \mathbf{7} & \mathbf{8} & \mathbf{9} \\ \mathbf{2} & \mathbf{1} & \mathbf{3} & \mathbf{4} & \mathbf{5} & \mathbf{6} & \mathbf{7} & \mathbf{9} & \mathbf{8} \\ \mathbf{3} & \mathbf{3} & \mathbf{1}+\mathbf{2}+\mathbf{7} & \mathbf{6}+\mathbf{7} & \mathbf{5}+\mathbf{6} & \mathbf{4}+\mathbf{5} & \mathbf{3}+\mathbf{4} & \mathbf{8}+\mathbf{9} & \mathbf{8}+\mathbf{9} \\ \mathbf{4} & \mathbf{4} & \mathbf{6}+\mathbf{7} & \mathbf{1}+\mathbf{2}+\mathbf{5} & \mathbf{4}+\mathbf{7} & \mathbf{3}+\mathbf{6} & \mathbf{3}+\mathbf{5} & \mathbf{8}+\mathbf{9} & \mathbf{8}+\mathbf{9} \\ \mathbf{5} & \mathbf{5} & \mathbf{5}+\mathbf{6} & \mathbf{4}+\mathbf{7} & \mathbf{1}+\mathbf{2}+\mathbf{3} & \mathbf{3}+\mathbf{7} & \mathbf{4}+\mathbf{6} & \mathbf{8}+\mathbf{9} & \mathbf{8}+\mathbf{9} \\ \mathbf{6} & \mathbf{6} & \mathbf{4}+\mathbf{5} & \mathbf{3}+\mathbf{6} & \mathbf{3}+\mathbf{7} & \mathbf{1}+\mathbf{2}+\mathbf{4} & \mathbf{5}+\mathbf{7} & \mathbf{8}+\mathbf{9} & \mathbf{8}+\mathbf{9} \\ \mathbf{7} & \mathbf{7} & \mathbf{3}+\mathbf{4} & \mathbf{3}+\mathbf{5} & \mathbf{4}+\mathbf{6} & \mathbf{5}+\mathbf{7} & \mathbf{1}+\mathbf{2}+\mathbf{6} & \mathbf{8}+\mathbf{9} & \mathbf{8}+\mathbf{9} \\ \mathbf{8} & \mathbf{9} & \mathbf{8}+\mathbf{9} & \mathbf{8}+\mathbf{9} & \mathbf{8}+\mathbf{9} & \mathbf{8}+\mathbf{9} & \mathbf{8}+\mathbf{9} & \mathbf{1}+\mathbf{3}+\mathbf{4}+\mathbf{5}+\mathbf{6}+\mathbf{7}+\mathbf{9} & \mathbf{2}+\mathbf{3}+\mathbf{4}+\mathbf{5}+\mathbf{6}+\mathbf{7}+\mathbf{8} \\ \mathbf{9} & \mathbf{8} & \mathbf{8}+\mathbf{9} & \mathbf{8}+\mathbf{9} & \mathbf{8}+\mathbf{9} & \mathbf{8}+\mathbf{9} & \mathbf{8}+\mathbf{9} & \mathbf{2}+\mathbf{3}+\mathbf{4}+\mathbf{5}+\mathbf{6}+\mathbf{7}+\mathbf{8} & \mathbf{1}+\mathbf{3}+\mathbf{4}+\mathbf{5}+\mathbf{6}+\mathbf{7}+\mathbf{9} \\ \hline \end{array}\]The fusion rules are invariant under the group generated by the following permutations:
\[\{(\mathbf{3} \ \mathbf{4} \ \mathbf{7} \ \mathbf{5} \ \mathbf{6}), (\mathbf{3} \ \mathbf{5} \ \mathbf{4} \ \mathbf{6} \ \mathbf{7})\}\]The following elements form non-trivial sub fusion rings
| Elements | SubRing |
|---|---|
| \(\{\mathbf{1},\mathbf{2}\}\) | \(\mathbb{Z}_2:\ \text{FR}^{2,0}_{1}\) |
| \(\{\mathbf{1},\mathbf{2},\mathbf{3},\mathbf{4},\mathbf{5},\mathbf{6},\mathbf{7}\}\) | \(\text{FR}^{7,0}_{6}\) |
Frobenius-Perron Dimensions
| Particle | Numeric | Symbolic |
|---|---|---|
| \(\mathbf{1}\) | \(1.\) | \(1\) |
| \(\mathbf{2}\) | \(1.\) | \(1\) |
| \(\mathbf{3}\) | \(2.\) | \(2\) |
| \(\mathbf{4}\) | \(2.\) | \(2\) |
| \(\mathbf{5}\) | \(2.\) | \(2\) |
| \(\mathbf{6}\) | \(2.\) | \(2\) |
| \(\mathbf{7}\) | \(2.\) | \(2\) |
| \(\mathbf{8}\) | \(3.8541\) | \(\frac{1}{2} \left(1+3 \sqrt{5}\right)\) |
| \(\mathbf{9}\) | \(3.8541\) | \(\frac{1}{2} \left(1+3 \sqrt{5}\right)\) |
| \(\mathcal{D}_{FP}^2\) | \(51.7082\) | \(22+\frac{1}{2} \left(1+3 \sqrt{5}\right)^2\) |
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{9} & \mathbf{8} \\ \hline 1 & 1 & 2 & 2 & 2 & 2 & 2 & \frac{1}{2} \left(1+3 \sqrt{5}\right) & \frac{1}{2} \left(1+3 \sqrt{5}\right) \\ 1 & 1 & 2 & 2 & 2 & 2 & 2 & \frac{1}{2} \left(1-3 \sqrt{5}\right) & \frac{1}{2} \left(1-3 \sqrt{5}\right) \\ 1 & 1 & \text{Root}\left[x^5+x^4-4 x^3-3 x^2+3 x+1,4\right] & \text{Root}\left[x^5+x^4-4 x^3-3 x^2+3 x+1,1\right] & \text{Root}\left[x^5+x^4-4 x^3-3 x^2+3 x+1,5\right] & \text{Root}\left[x^5+x^4-4 x^3-3 x^2+3 x+1,3\right] & \text{Root}\left[x^5+x^4-4 x^3-3 x^2+3 x+1,2\right] & 0 & 0 \\ 1 & 1 & \text{Root}\left[x^5+x^4-4 x^3-3 x^2+3 x+1,3\right] & \text{Root}\left[x^5+x^4-4 x^3-3 x^2+3 x+1,4\right] & \text{Root}\left[x^5+x^4-4 x^3-3 x^2+3 x+1,2\right] & \text{Root}\left[x^5+x^4-4 x^3-3 x^2+3 x+1,5\right] & \text{Root}\left[x^5+x^4-4 x^3-3 x^2+3 x+1,1\right] & 0 & 0 \\ 1 & 1 & \text{Root}\left[x^5+x^4-4 x^3-3 x^2+3 x+1,5\right] & \text{Root}\left[x^5+x^4-4 x^3-3 x^2+3 x+1,3\right] & \text{Root}\left[x^5+x^4-4 x^3-3 x^2+3 x+1,1\right] & \text{Root}\left[x^5+x^4-4 x^3-3 x^2+3 x+1,2\right] & \text{Root}\left[x^5+x^4-4 x^3-3 x^2+3 x+1,4\right] & 0 & 0 \\ 1 & 1 & \text{Root}\left[x^5+x^4-4 x^3-3 x^2+3 x+1,1\right] & \text{Root}\left[x^5+x^4-4 x^3-3 x^2+3 x+1,2\right] & \text{Root}\left[x^5+x^4-4 x^3-3 x^2+3 x+1,3\right] & \text{Root}\left[x^5+x^4-4 x^3-3 x^2+3 x+1,4\right] & \text{Root}\left[x^5+x^4-4 x^3-3 x^2+3 x+1,5\right] & 0 & 0 \\ 1 & 1 & \text{Root}\left[x^5+x^4-4 x^3-3 x^2+3 x+1,2\right] & \text{Root}\left[x^5+x^4-4 x^3-3 x^2+3 x+1,5\right] & \text{Root}\left[x^5+x^4-4 x^3-3 x^2+3 x+1,4\right] & \text{Root}\left[x^5+x^4-4 x^3-3 x^2+3 x+1,1\right] & \text{Root}\left[x^5+x^4-4 x^3-3 x^2+3 x+1,3\right] & 0 & 0 \\ 1 & -1 & 0 & 0 & 0 & 0 & 0 & \frac{1}{2} \left(1+\sqrt{5}\right) & \frac{1}{2} \left(-1-\sqrt{5}\right) \\ 1 & -1 & 0 & 0 & 0 & 0 & 0 & \frac{1}{2} \left(1-\sqrt{5}\right) & \frac{1}{2} \left(\sqrt{5}-1\right) \\ \hline \end{array}\]The numeric character table is the following
\[\begin{array}{|rrrrrrrrr|} \hline \mathbf{1} & \mathbf{2} & \mathbf{3} & \mathbf{4} & \mathbf{5} & \mathbf{6} & \mathbf{7} & \mathbf{9} & \mathbf{8} \\ \hline 1.000 & 1.000 & 2.000 & 2.000 & 2.000 & 2.000 & 2.000 & 3.854 & 3.854 \\ 1.000 & 1.000 & 2.000 & 2.000 & 2.000 & 2.000 & 2.000 & -2.854 & -2.854 \\ 1.000 & 1.000 & 0.8308 & -1.919 & 1.683 & -0.2846 & -1.310 & 0 & 0 \\ 1.000 & 1.000 & -0.2846 & 0.8308 & -1.310 & 1.683 & -1.919 & 0 & 0 \\ 1.000 & 1.000 & 1.683 & -0.2846 & -1.919 & -1.310 & 0.8308 & 0 & 0 \\ 1.000 & 1.000 & -1.919 & -1.310 & -0.2846 & 0.8308 & 1.683 & 0 & 0 \\ 1.000 & 1.000 & -1.310 & 1.683 & 0.8308 & -1.919 & -0.2846 & 0 & 0 \\ 1.000 & -1.000 & 0 & 0 & 0 & 0 & 0 & 1.618 & -1.618 \\ 1.000 & -1.000 & 0 & 0 & 0 & 0 & 0 & -0.6180 & 0.6180 \\ \hline \end{array}\]Representations of $SL_2(\mathbb{Z})$
This fusion ring does not provide any representations of $SL_2(\mathbb{Z}).$
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
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