\(\text{FR}^{6,0}_{19}\)
Fusion Rules
\[\begin{array}{|llllll|} \hline \mathbf{1} & \mathbf{2} & \mathbf{3} & \mathbf{4} & \mathbf{5} & \mathbf{6} \\ \mathbf{2} & \mathbf{1}+\mathbf{3}+\mathbf{4}+\mathbf{5} & \mathbf{2}+\mathbf{3}+\mathbf{6} & \mathbf{2}+\mathbf{4}+\mathbf{6} & \mathbf{2}+\mathbf{5}+\mathbf{6} & \mathbf{3}+\mathbf{4}+\mathbf{5}+\mathbf{6} \\ \mathbf{3} & \mathbf{2}+\mathbf{3}+\mathbf{6} & \mathbf{1}+\mathbf{2}+\mathbf{4}+\mathbf{5} & \mathbf{3}+\mathbf{4}+\mathbf{6} & \mathbf{3}+\mathbf{5}+\mathbf{6} & \mathbf{2}+\mathbf{4}+\mathbf{5}+\mathbf{6} \\ \mathbf{4} & \mathbf{2}+\mathbf{4}+\mathbf{6} & \mathbf{3}+\mathbf{4}+\mathbf{6} & \mathbf{1}+\mathbf{2}+\mathbf{3}+\mathbf{5} & \mathbf{4}+\mathbf{5}+\mathbf{6} & \mathbf{2}+\mathbf{3}+\mathbf{5}+\mathbf{6} \\ \mathbf{5} & \mathbf{2}+\mathbf{5}+\mathbf{6} & \mathbf{3}+\mathbf{5}+\mathbf{6} & \mathbf{4}+\mathbf{5}+\mathbf{6} & \mathbf{1}+\mathbf{2}+\mathbf{3}+\mathbf{4} & \mathbf{2}+\mathbf{3}+\mathbf{4}+\mathbf{6} \\ \mathbf{6} & \mathbf{3}+\mathbf{4}+\mathbf{5}+\mathbf{6} & \mathbf{2}+\mathbf{4}+\mathbf{5}+\mathbf{6} & \mathbf{2}+\mathbf{3}+\mathbf{5}+\mathbf{6} & \mathbf{2}+\mathbf{3}+\mathbf{4}+\mathbf{6} & \mathbf{1}+\mathbf{2}+\mathbf{3}+\mathbf{4}+\mathbf{5}+\mathbf{6} \\ \hline \end{array}\]The fusion rules are invariant under the group generated by the following permutations:
\[\{(\mathbf{2} \ \mathbf{3}), (\mathbf{2} \ \mathbf{4}), (\mathbf{2} \ \mathbf{5}), (\mathbf{3} \ \mathbf{4}), (\mathbf{3} \ \mathbf{5}), (\mathbf{4} \ \mathbf{5})\}\]Quantum Dimensions
| Particle | Numeric | Symbolic |
|---|---|---|
| \(\mathbf{1}\) | \(1.\) | \(1\) |
| \(\mathbf{2}\) | \(3.30278\) | \(\frac{1}{2} \left(3+\sqrt{13}\right)\) |
| \(\mathbf{3}\) | \(3.30278\) | \(\frac{1}{2} \left(3+\sqrt{13}\right)\) |
| \(\mathbf{4}\) | \(3.30278\) | \(\frac{1}{2} \left(3+\sqrt{13}\right)\) |
| \(\mathbf{5}\) | \(3.30278\) | \(\frac{1}{2} \left(3+\sqrt{13}\right)\) |
| \(\mathbf{6}\) | \(4.30278\) | \(\frac{1}{2} \left(5+\sqrt{13}\right)\) |
| \(\mathcal{D}_{FP}^2\) | \(63.1472\) | \(1+\left(3+\sqrt{13}\right)^2+\frac{1}{4} \left(5+\sqrt{13}\right)^2\) |
Characters
The symbolic character table is the following
\[\begin{array}{|cccccc|} \hline \mathbf{1} & \mathbf{2} & \mathbf{3} & \mathbf{4} & \mathbf{5} & \mathbf{6} \\ \hline 1 & \frac{1}{2} \left(3+\sqrt{13}\right) & \frac{1}{2} \left(3+\sqrt{13}\right) & \frac{1}{2} \left(3+\sqrt{13}\right) & \frac{1}{2} \left(3+\sqrt{13}\right) & \frac{1}{2} \left(5+\sqrt{13}\right) \\ 1 & 1 & 1 & 1 & -2 & -1 \\ 1 & 1 & 1 & -2 & 1 & -1 \\ 1 & 1 & -2 & 1 & 1 & -1 \\ 1 & -2 & 1 & 1 & 1 & -1 \\ 1 & \frac{1}{2} \left(3-\sqrt{13}\right) & \frac{1}{2} \left(3-\sqrt{13}\right) & \frac{1}{2} \left(3-\sqrt{13}\right) & \frac{1}{2} \left(3-\sqrt{13}\right) & \frac{1}{2} \left(5-\sqrt{13}\right) \\ \hline \end{array}\]The numeric character table is the following
\[\begin{array}{|rrrrrr|} \hline \mathbf{1} & \mathbf{2} & \mathbf{3} & \mathbf{4} & \mathbf{5} & \mathbf{6} \\ \hline 1.000 & 3.303 & 3.303 & 3.303 & 3.303 & 4.303 \\ 1.000 & 1.000 & 1.000 & 1.000 & -2.000 & -1.000 \\ 1.000 & 1.000 & 1.000 & -2.000 & 1.000 & -1.000 \\ 1.000 & 1.000 & -2.000 & 1.000 & 1.000 & -1.000 \\ 1.000 & -2.000 & 1.000 & 1.000 & 1.000 & -1.000 \\ 1.000 & -0.3028 & -0.3028 & -0.3028 & -0.3028 & 0.6972 \\ \hline \end{array}\]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 unitary categorifications because of the Commutative Schur product criterion.
Data
Download links for numeric data: