\(\text{FR}^{9,2}_{43}\)
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{3}+\mathbf{4} & \mathbf{3}+\mathbf{4}+\mathbf{5} & \mathbf{4}+\mathbf{5}+\mathbf{6} & \mathbf{5}+\mathbf{6}+\mathbf{7} & \mathbf{6}+\mathbf{8}+\mathbf{9} & \mathbf{7}+\mathbf{8}+\mathbf{9} & \mathbf{7}+\mathbf{8}+\mathbf{9} \\ \mathbf{4} & \mathbf{4} & \mathbf{3}+\mathbf{4}+\mathbf{5} & \mathbf{1}+\mathbf{2}+\mathbf{3}+\mathbf{5}+\mathbf{6} & \mathbf{3}+\mathbf{4}+\mathbf{6}+\mathbf{7} & \mathbf{4}+\mathbf{5}+\mathbf{8}+\mathbf{9} & \mathbf{5}+\mathbf{7}+\mathbf{8}+\mathbf{9} & \mathbf{6}+\mathbf{7}+\mathbf{8}+\mathbf{9} & \mathbf{6}+\mathbf{7}+\mathbf{8}+\mathbf{9} \\ \mathbf{5} & \mathbf{5} & \mathbf{4}+\mathbf{5}+\mathbf{6} & \mathbf{3}+\mathbf{4}+\mathbf{6}+\mathbf{7} & \mathbf{1}+\mathbf{2}+\mathbf{3}+\mathbf{5}+\mathbf{8}+\mathbf{9} & \mathbf{3}+\mathbf{4}+\mathbf{7}+\mathbf{8}+\mathbf{9} & \mathbf{4}+\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{6} & \mathbf{6} & \mathbf{5}+\mathbf{6}+\mathbf{7} & \mathbf{4}+\mathbf{5}+\mathbf{8}+\mathbf{9} & \mathbf{3}+\mathbf{4}+\mathbf{7}+\mathbf{8}+\mathbf{9} & \mathbf{1}+\mathbf{2}+\mathbf{3}+\mathbf{6}+\mathbf{7}+\mathbf{8}+\mathbf{9} & \mathbf{3}+\mathbf{5}+\mathbf{6}+\mathbf{7}+\mathbf{8}+\mathbf{9} & \mathbf{4}+\mathbf{5}+\mathbf{6}+\mathbf{7}+\mathbf{8}+\mathbf{9} & \mathbf{4}+\mathbf{5}+\mathbf{6}+\mathbf{7}+\mathbf{8}+\mathbf{9} \\ \mathbf{7} & \mathbf{7} & \mathbf{6}+\mathbf{8}+\mathbf{9} & \mathbf{5}+\mathbf{7}+\mathbf{8}+\mathbf{9} & \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{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{8} & \mathbf{9} & \mathbf{7}+\mathbf{8}+\mathbf{9} & \mathbf{6}+\mathbf{7}+\mathbf{8}+\mathbf{9} & \mathbf{5}+\mathbf{6}+\mathbf{7}+\mathbf{8}+\mathbf{9} & \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{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} \\ \mathbf{9} & \mathbf{8} & \mathbf{7}+\mathbf{8}+\mathbf{9} & \mathbf{6}+\mathbf{7}+\mathbf{8}+\mathbf{9} & \mathbf{5}+\mathbf{6}+\mathbf{7}+\mathbf{8}+\mathbf{9} & \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{9} & \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:
\[\{(\mathbf{8} \ \mathbf{9})\}\]The following particles form non-trivial sub fusion rings
| Particles | SubRing |
|---|---|
| \(\{\mathbf{1},\mathbf{2}\}\) | \(\mathbb{Z}_2:\ \text{FR}^{2,0}_{1}\) |
Quantum Dimensions
| Particle | Numeric | Symbolic |
|---|---|---|
| \(\mathbf{1}\) | \(1.\) | \(1\) |
| \(\mathbf{2}\) | \(1.\) | \(1\) |
| \(\mathbf{3}\) | \(2.97179\) | \(\text{Root}\left[x^6-4 x^5-2 x^4+15 x^3+4 x^2-11 x-2,6\right]\) |
| \(\mathbf{4}\) | \(3.85976\) | \(\text{Root}\left[x^6-4 x^5-7 x^4+29 x^3+11 x^2-45 x+16,6\right]\) |
| \(\mathbf{5}\) | \(4.63885\) | \(\text{Root}\left[x^6-7 x^5+9 x^4+13 x^3-16 x^2-11 x+2,6\right]\) |
| \(\mathbf{6}\) | \(5.2871\) | \(\text{Root}\left[x^6-7 x^5+7 x^4+13 x^3-10 x^2-7 x+2,6\right]\) |
| \(\mathbf{7}\) | \(5.78621\) | \(\text{Root}\left[x^6-4 x^5-13 x^4+10 x^3+32 x^2-x-16,6\right]\) |
| \(\mathbf{8}\) | \(5.95416\) | \(\text{Root}\left[x^6-4 x^5-12 x^4+x^3+7 x^2-1,6\right]\) |
| \(\mathbf{9}\) | \(5.95416\) | \(\text{Root}\left[x^6-4 x^5-12 x^4+x^3+7 x^2-1,6\right]\) |
| \(\mathcal{D}_{FP}^2\) | \(179.586\) | \(\text{Root}\left[x^6-4 x^5-2 x^4+15 x^3+4 x^2-11 x-2,6\right]^2+\text{Root}\left[x^6-4 x^5-7 x^4+29 x^3+11 x^2-45 x+16,6\right]^2+\text{Root}\left[x^6-7 x^5+9 x^4+13 x^3-16 x^2-11 x+2,6\right]^2+\text{Root}\left[x^6-7 x^5+7 x^4+13 x^3-10 x^2-7 x+2,6\right]^2+\text{Root}\left[x^6-4 x^5-13 x^4+10 x^3+32 x^2-x-16,6\right]^2+2 \text{Root}\left[x^6-4 x^5-12 x^4+x^3+7 x^2-1,6\right]^2+2\) |
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 & \text{Root}\left[x^6-4 x^5-2 x^4+15 x^3+4 x^2-11 x-2,6\right] & \text{Root}\left[x^6-4 x^5-7 x^4+29 x^3+11 x^2-45 x+16,6\right] & \text{Root}\left[x^6-7 x^5+9 x^4+13 x^3-16 x^2-11 x+2,6\right] & \text{Root}\left[x^6-7 x^5+7 x^4+13 x^3-10 x^2-7 x+2,6\right] & \text{Root}\left[x^6-4 x^5-13 x^4+10 x^3+32 x^2-x-16,6\right] & \text{Root}\left[x^6-4 x^5-12 x^4+x^3+7 x^2-1,6\right] & \text{Root}\left[x^6-4 x^5-12 x^4+x^3+7 x^2-1,6\right] \\ 1 & 1 & \text{Root}\left[x^6-4 x^5-2 x^4+15 x^3+4 x^2-11 x-2,5\right] & \text{Root}\left[x^6-4 x^5-7 x^4+29 x^3+11 x^2-45 x+16,5\right] & \text{Root}\left[x^6-7 x^5+9 x^4+13 x^3-16 x^2-11 x+2,4\right] & \text{Root}\left[x^6-7 x^5+7 x^4+13 x^3-10 x^2-7 x+2,3\right] & \text{Root}\left[x^6-4 x^5-13 x^4+10 x^3+32 x^2-x-16,2\right] & \text{Root}\left[x^6-4 x^5-12 x^4+x^3+7 x^2-1,1\right] & \text{Root}\left[x^6-4 x^5-12 x^4+x^3+7 x^2-1,1\right] \\ 1 & 1 & 2 & 0 & -2 & -2 & 0 & 1 & 1 \\ 1 & 1 & \text{Root}\left[x^6-4 x^5-2 x^4+15 x^3+4 x^2-11 x-2,4\right] & \text{Root}\left[x^6-4 x^5-7 x^4+29 x^3+11 x^2-45 x+16,1\right] & \text{Root}\left[x^6-7 x^5+9 x^4+13 x^3-16 x^2-11 x+2,2\right] & \text{Root}\left[x^6-7 x^5+7 x^4+13 x^3-10 x^2-7 x+2,5\right] & \text{Root}\left[x^6-4 x^5-13 x^4+10 x^3+32 x^2-x-16,4\right] & \text{Root}\left[x^6-4 x^5-12 x^4+x^3+7 x^2-1,2\right] & \text{Root}\left[x^6-4 x^5-12 x^4+x^3+7 x^2-1,2\right] \\ 1 & 1 & \text{Root}\left[x^6-4 x^5-2 x^4+15 x^3+4 x^2-11 x-2,3\right] & \text{Root}\left[x^6-4 x^5-7 x^4+29 x^3+11 x^2-45 x+16,2\right] & \text{Root}\left[x^6-7 x^5+9 x^4+13 x^3-16 x^2-11 x+2,5\right] & \text{Root}\left[x^6-7 x^5+7 x^4+13 x^3-10 x^2-7 x+2,1\right] & \text{Root}\left[x^6-4 x^5-13 x^4+10 x^3+32 x^2-x-16,3\right] & \text{Root}\left[x^6-4 x^5-12 x^4+x^3+7 x^2-1,5\right] & \text{Root}\left[x^6-4 x^5-12 x^4+x^3+7 x^2-1,5\right] \\ 1 & 1 & \text{Root}\left[x^6-4 x^5-2 x^4+15 x^3+4 x^2-11 x-2,2\right] & \text{Root}\left[x^6-4 x^5-7 x^4+29 x^3+11 x^2-45 x+16,3\right] & \text{Root}\left[x^6-7 x^5+9 x^4+13 x^3-16 x^2-11 x+2,3\right] & \text{Root}\left[x^6-7 x^5+7 x^4+13 x^3-10 x^2-7 x+2,2\right] & \text{Root}\left[x^6-4 x^5-13 x^4+10 x^3+32 x^2-x-16,5\right] & \text{Root}\left[x^6-4 x^5-12 x^4+x^3+7 x^2-1,3\right] & \text{Root}\left[x^6-4 x^5-12 x^4+x^3+7 x^2-1,3\right] \\ 1 & 1 & \text{Root}\left[x^6-4 x^5-2 x^4+15 x^3+4 x^2-11 x-2,1\right] & \text{Root}\left[x^6-4 x^5-7 x^4+29 x^3+11 x^2-45 x+16,4\right] & \text{Root}\left[x^6-7 x^5+9 x^4+13 x^3-16 x^2-11 x+2,1\right] & \text{Root}\left[x^6-7 x^5+7 x^4+13 x^3-10 x^2-7 x+2,4\right] & \text{Root}\left[x^6-4 x^5-13 x^4+10 x^3+32 x^2-x-16,1\right] & \text{Root}\left[x^6-4 x^5-12 x^4+x^3+7 x^2-1,4\right] & \text{Root}\left[x^6-4 x^5-12 x^4+x^3+7 x^2-1,4\right] \\ 1 & -1 & 0 & 0 & 0 & 0 & 0 & i & -i \\ 1 & -1 & 0 & 0 & 0 & 0 & 0 & -i & i \\ \hline \end{array}\]The numeric character table is the following
\[\begin{array}{|rrrrrrrrr|} \hline \mathbf{1} & \mathbf{2} & \mathbf{3} & \mathbf{4} & \mathbf{5} & \mathbf{6} & \mathbf{7} & \mathbf{9} & \mathbf{8} \\ \hline 1.000 & 1.000 & 2.972 & 3.860 & 4.639 & 5.287 & 5.786 & 5.954 & 5.954 \\ 1.000 & 1.000 & 2.694 & 2.565 & 1.652 & 0.2337 & -1.256 & -1.809 & -1.809 \\ 1.000 & 1.000 & 2.000 & 0 & -2.000 & -2.000 & 0 & 1.000 & 1.000 \\ 1.000 & 1.000 & 0.9483 & -2.049 & -0.8424 & 2.093 & 0.7342 & -0.6981 & -0.6981 \\ 1.000 & 1.000 & -0.1781 & -1.790 & 2.287 & -0.9043 & -1.222 & 0.5609 & 0.5609 \\ 1.000 & 1.000 & -1.147 & 0.4632 & 0.1526 & -0.7908 & 1.545 & -0.4910 & -0.4910 \\ 1.000 & 1.000 & -1.289 & 0.9511 & -0.8880 & 1.082 & -1.588 & 0.4829 & 0.4829 \\ 1.000 & -1.000 & 0 & 0 & 0 & 0 & 0 & 1.000 i & -1.000 i \\ 1.000 & -1.000 & 0 & 0 & 0 & 0 & 0 & -1.000 i & 1.000 i \\ \hline \end{array}\]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 zero spectrum criterion.
Data
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