\(\text{FR}^{9,6}_{11}\)
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{3} & \mathbf{1} & \mathbf{6} & \mathbf{4} & \mathbf{5} & \mathbf{8} & \mathbf{9} & \mathbf{7} \\ \mathbf{3} & \mathbf{1} & \mathbf{2} & \mathbf{5} & \mathbf{6} & \mathbf{4} & \mathbf{9} & \mathbf{7} & \mathbf{8} \\ \mathbf{4} & \mathbf{6} & \mathbf{5} & \mathbf{1}+\mathbf{8}+\mathbf{9} & \mathbf{3}+\mathbf{7}+\mathbf{8} & \mathbf{2}+\mathbf{7}+\mathbf{9} & \mathbf{5}+\mathbf{6}+\mathbf{7} & \mathbf{4}+\mathbf{5}+\mathbf{8} & \mathbf{4}+\mathbf{6}+\mathbf{9} \\ \mathbf{5} & \mathbf{4} & \mathbf{6} & \mathbf{3}+\mathbf{7}+\mathbf{8} & \mathbf{2}+\mathbf{7}+\mathbf{9} & \mathbf{1}+\mathbf{8}+\mathbf{9} & \mathbf{4}+\mathbf{6}+\mathbf{9} & \mathbf{5}+\mathbf{6}+\mathbf{7} & \mathbf{4}+\mathbf{5}+\mathbf{8} \\ \mathbf{6} & \mathbf{5} & \mathbf{4} & \mathbf{2}+\mathbf{7}+\mathbf{9} & \mathbf{1}+\mathbf{8}+\mathbf{9} & \mathbf{3}+\mathbf{7}+\mathbf{8} & \mathbf{4}+\mathbf{5}+\mathbf{8} & \mathbf{4}+\mathbf{6}+\mathbf{9} & \mathbf{5}+\mathbf{6}+\mathbf{7} \\ \mathbf{7} & \mathbf{8} & \mathbf{9} & \mathbf{5}+\mathbf{6}+\mathbf{7} & \mathbf{4}+\mathbf{6}+\mathbf{9} & \mathbf{4}+\mathbf{5}+\mathbf{8} & \mathbf{1}+\mathbf{4}+\mathbf{8}+\mathbf{9} & \mathbf{2}+\mathbf{6}+\mathbf{7}+\mathbf{9} & \mathbf{3}+\mathbf{5}+\mathbf{7}+\mathbf{8} \\ \mathbf{8} & \mathbf{9} & \mathbf{7} & \mathbf{4}+\mathbf{5}+\mathbf{8} & \mathbf{5}+\mathbf{6}+\mathbf{7} & \mathbf{4}+\mathbf{6}+\mathbf{9} & \mathbf{2}+\mathbf{6}+\mathbf{7}+\mathbf{9} & \mathbf{3}+\mathbf{5}+\mathbf{7}+\mathbf{8} & \mathbf{1}+\mathbf{4}+\mathbf{8}+\mathbf{9} \\ \mathbf{9} & \mathbf{7} & \mathbf{8} & \mathbf{4}+\mathbf{6}+\mathbf{9} & \mathbf{4}+\mathbf{5}+\mathbf{8} & \mathbf{5}+\mathbf{6}+\mathbf{7} & \mathbf{3}+\mathbf{5}+\mathbf{7}+\mathbf{8} & \mathbf{1}+\mathbf{4}+\mathbf{8}+\mathbf{9} & \mathbf{2}+\mathbf{6}+\mathbf{7}+\mathbf{9} \\ \hline \end{array}\]The fusion rules are invariant under the group generated by the following permutations:
\[\{(\mathbf{2} \ \mathbf{3}) (\mathbf{5} \ \mathbf{6}) (\mathbf{8} \ \mathbf{9})\}\]The following particles form non-trivial sub fusion rings
| Particles | SubRing |
|---|---|
| \(\{\mathbf{1},\mathbf{2},\mathbf{3}\}\) | \(\mathbb{Z}_3:\ \text{FR}^{3,2}_{1}\) |
Quantum Dimensions
| Particle | Numeric | Symbolic |
|---|---|---|
| \(\mathbf{1}\) | \(1.\) | \(1\) |
| \(\mathbf{2}\) | \(1.\) | \(1\) |
| \(\mathbf{3}\) | \(1.\) | \(1\) |
| \(\mathbf{4}\) | \(2.70928\) | \(\text{Root}\left[x^3-x^2-5 x+1,3\right]\) |
| \(\mathbf{5}\) | \(2.70928\) | \(\text{Root}\left[x^3-x^2-5 x+1,3\right]\) |
| \(\mathbf{6}\) | \(2.70928\) | \(\text{Root}\left[x^3-x^2-5 x+1,3\right]\) |
| \(\mathbf{7}\) | \(3.17009\) | \(\text{Root}\left[x^3-4 x^2+2 x+2,3\right]\) |
| \(\mathbf{8}\) | \(3.17009\) | \(\text{Root}\left[x^3-4 x^2+2 x+2,3\right]\) |
| \(\mathbf{9}\) | \(3.17009\) | \(\text{Root}\left[x^3-4 x^2+2 x+2,3\right]\) |
| \(\mathcal{D}_{FP}^2\) | \(55.1689\) | \(3 \text{Root}\left[x^3-x^2-5 x+1,3\right]^2+3 \text{Root}\left[x^3-4 x^2+2 x+2,3\right]^2+3\) |
Characters
The symbolic character table is the following
\[\begin{array}{|ccccccccc|} \hline \mathbf{1} & \mathbf{2} & \mathbf{3} & \mathbf{4} & \mathbf{5} & \mathbf{6} & \mathbf{7} & \mathbf{9} & \mathbf{8} \\ \hline 1. & 1. & 1. & 2.70928 & 2.70928 & 2.70928 & 3.17009 & 3.17009 & 3.17009 \\ 1. & 1. & 1. & 0.193937 & 0.193937 & 0.193937 & -0.481194 & -0.481194 & -0.481194 \\ 1. & 1. & 1. & -1.90321 & -1.90321 & -1.90321 & 1.31111 & 1.31111 & 1.31111 \\ 1. & -0.5+0.866025 i & -0.5-0.866025 i & 0.445042\, +0. i & -0.222521-0.385418 i & -0.222521+0.385418 i & 0.801938\, +0. i & -0.400969-0.694498 i & -0.400969+0.694498 i \\ 1. & -0.5-0.866025 i & -0.5+0.866025 i & 0.445042\, +0. i & -0.222521+0.385418 i & -0.222521-0.385418 i & 0.801938\, +0. i & -0.400969+0.694498 i & -0.400969-0.694498 i \\ 1. & -0.5+0.866025 i & -0.5-0.866025 i & 1.80194\, +0. i & -0.900969-1.56052 i & -0.900969+1.56052 i & -2.24698 & 1.12349\, +1.94594 i & 1.12349\, -1.94594 i \\ 1. & -0.5-0.866025 i & -0.5+0.866025 i & 1.80194\, +0. i & -0.900969+1.56052 i & -0.900969-1.56052 i & -2.24698 & 1.12349\, -1.94594 i & 1.12349\, +1.94594 i \\ 1.\, +0. i & -0.5+0.866025 i & -0.5-0.866025 i & -1.24698+0. i & 0.62349\, +1.07992 i & 0.62349\, -1.07992 i & -0.554958+0. i & 0.277479\, +0.480608 i & 0.277479\, -0.480608 i \\ 1.\, +0. i & -0.5-0.866025 i & -0.5+0.866025 i & -1.24698+0. i & 0.62349\, -1.07992 i & 0.62349\, +1.07992 i & -0.554958+0. i & 0.277479\, -0.480608 i & 0.277479\, +0.480608 i \\ \hline \end{array}\]The numeric character table is the following
\[\begin{array}{|rrrrrrrrr|} \hline \mathbf{1} & \mathbf{2} & \mathbf{3} & \mathbf{4} & \mathbf{5} & \mathbf{6} & \mathbf{7} & \mathbf{9} & \mathbf{8} \\ \hline 1. & 1. & 1. & 2.70928 & 2.70928 & 2.70928 & 3.17009 & 3.17009 & 3.17009 \\ 1. & 1. & 1. & 0.193937 & 0.193937 & 0.193937 & -0.481194 & -0.481194 & -0.481194 \\ 1. & 1. & 1. & -1.90321 & -1.90321 & -1.90321 & 1.31111 & 1.31111 & 1.31111 \\ 1. & -0.5+0.866025 i & -0.5-0.866025 i & 0.445042\, +0. i & -0.222521-0.385418 i & -0.222521+0.385418 i & 0.801938\, +0. i & -0.400969-0.694498 i & -0.400969+0.694498 i \\ 1. & -0.5-0.866025 i & -0.5+0.866025 i & 0.445042\, +0. i & -0.222521+0.385418 i & -0.222521-0.385418 i & 0.801938\, +0. i & -0.400969+0.694498 i & -0.400969-0.694498 i \\ 1. & -0.5+0.866025 i & -0.5-0.866025 i & 1.80194\, +0. i & -0.900969-1.56052 i & -0.900969+1.56052 i & -2.24698 & 1.12349\, +1.94594 i & 1.12349\, -1.94594 i \\ 1. & -0.5-0.866025 i & -0.5+0.866025 i & 1.80194\, +0. i & -0.900969+1.56052 i & -0.900969-1.56052 i & -2.24698 & 1.12349\, -1.94594 i & 1.12349\, +1.94594 i \\ 1.\, +0. i & -0.5+0.866025 i & -0.5-0.866025 i & -1.24698+0. i & 0.62349\, +1.07992 i & 0.62349\, -1.07992 i & -0.554958+0. i & 0.277479\, +0.480608 i & 0.277479\, -0.480608 i \\ 1.\, +0. i & -0.5-0.866025 i & -0.5+0.866025 i & -1.24698+0. i & 0.62349\, -1.07992 i & 0.62349\, +1.07992 i & -0.554958+0. i & 0.277479\, -0.480608 i & 0.277479\, +0.480608 i \\ \hline \end{array}\]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 extended cyclotomic criterion.
Data
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