\(\text{FR}^{8,0}_{38}\)
Fusion Rules
\[\begin{array}{|llllllll|} \hline \mathbf{1} & \mathbf{2} & \mathbf{3} & \mathbf{4} & \mathbf{5} & \mathbf{6} & \mathbf{7} & \mathbf{8} \\ \mathbf{2} & \mathbf{1}+\mathbf{2}+\mathbf{4}+\mathbf{5}+\mathbf{6}+\mathbf{7} & \mathbf{3}+\mathbf{5}+\mathbf{6}+\mathbf{7}+\mathbf{8} & \mathbf{2}+\mathbf{4}+\mathbf{6}+\mathbf{7}+\mathbf{8} & \mathbf{2}+\mathbf{3}+\mathbf{5}+\mathbf{7}+\mathbf{8} & \mathbf{2}+\mathbf{3}+\mathbf{4}+\mathbf{6}+\mathbf{8} & \mathbf{2}+\mathbf{3}+\mathbf{4}+\mathbf{5}+\mathbf{8} & \mathbf{3}+\mathbf{4}+\mathbf{5}+\mathbf{6}+\mathbf{7}+\mathbf{8} \\ \mathbf{3} & \mathbf{3}+\mathbf{5}+\mathbf{6}+\mathbf{7}+\mathbf{8} & \mathbf{1}+\mathbf{2}+\mathbf{3}+\mathbf{4}+\mathbf{6}+\mathbf{7} & \mathbf{3}+\mathbf{4}+\mathbf{5}+\mathbf{7}+\mathbf{8} & \mathbf{2}+\mathbf{4}+\mathbf{5}+\mathbf{6}+\mathbf{8} & \mathbf{2}+\mathbf{3}+\mathbf{5}+\mathbf{6}+\mathbf{8} & \mathbf{2}+\mathbf{3}+\mathbf{4}+\mathbf{7}+\mathbf{8} & \mathbf{2}+\mathbf{4}+\mathbf{5}+\mathbf{6}+\mathbf{7}+\mathbf{8} \\ \mathbf{4} & \mathbf{2}+\mathbf{4}+\mathbf{6}+\mathbf{7}+\mathbf{8} & \mathbf{3}+\mathbf{4}+\mathbf{5}+\mathbf{7}+\mathbf{8} & \mathbf{1}+\mathbf{2}+\mathbf{3}+\mathbf{4}+\mathbf{5}+\mathbf{6} & \mathbf{3}+\mathbf{4}+\mathbf{5}+\mathbf{6}+\mathbf{8} & \mathbf{2}+\mathbf{4}+\mathbf{5}+\mathbf{7}+\mathbf{8} & \mathbf{2}+\mathbf{3}+\mathbf{6}+\mathbf{7}+\mathbf{8} & \mathbf{2}+\mathbf{3}+\mathbf{5}+\mathbf{6}+\mathbf{7}+\mathbf{8} \\ \mathbf{5} & \mathbf{2}+\mathbf{3}+\mathbf{5}+\mathbf{7}+\mathbf{8} & \mathbf{2}+\mathbf{4}+\mathbf{5}+\mathbf{6}+\mathbf{8} & \mathbf{3}+\mathbf{4}+\mathbf{5}+\mathbf{6}+\mathbf{8} & \mathbf{1}+\mathbf{2}+\mathbf{3}+\mathbf{4}+\mathbf{5}+\mathbf{7} & \mathbf{3}+\mathbf{4}+\mathbf{6}+\mathbf{7}+\mathbf{8} & \mathbf{2}+\mathbf{5}+\mathbf{6}+\mathbf{7}+\mathbf{8} & \mathbf{2}+\mathbf{3}+\mathbf{4}+\mathbf{6}+\mathbf{7}+\mathbf{8} \\ \mathbf{6} & \mathbf{2}+\mathbf{3}+\mathbf{4}+\mathbf{6}+\mathbf{8} & \mathbf{2}+\mathbf{3}+\mathbf{5}+\mathbf{6}+\mathbf{8} & \mathbf{2}+\mathbf{4}+\mathbf{5}+\mathbf{7}+\mathbf{8} & \mathbf{3}+\mathbf{4}+\mathbf{6}+\mathbf{7}+\mathbf{8} & \mathbf{1}+\mathbf{2}+\mathbf{3}+\mathbf{5}+\mathbf{6}+\mathbf{7} & \mathbf{4}+\mathbf{5}+\mathbf{6}+\mathbf{7}+\mathbf{8} & \mathbf{2}+\mathbf{3}+\mathbf{4}+\mathbf{5}+\mathbf{7}+\mathbf{8} \\ \mathbf{7} & \mathbf{2}+\mathbf{3}+\mathbf{4}+\mathbf{5}+\mathbf{8} & \mathbf{2}+\mathbf{3}+\mathbf{4}+\mathbf{7}+\mathbf{8} & \mathbf{2}+\mathbf{3}+\mathbf{6}+\mathbf{7}+\mathbf{8} & \mathbf{2}+\mathbf{5}+\mathbf{6}+\mathbf{7}+\mathbf{8} & \mathbf{4}+\mathbf{5}+\mathbf{6}+\mathbf{7}+\mathbf{8} & \mathbf{1}+\mathbf{3}+\mathbf{4}+\mathbf{5}+\mathbf{6}+\mathbf{7} & \mathbf{2}+\mathbf{3}+\mathbf{4}+\mathbf{5}+\mathbf{6}+\mathbf{8} \\ \mathbf{8} & \mathbf{3}+\mathbf{4}+\mathbf{5}+\mathbf{6}+\mathbf{7}+\mathbf{8} & \mathbf{2}+\mathbf{4}+\mathbf{5}+\mathbf{6}+\mathbf{7}+\mathbf{8} & \mathbf{2}+\mathbf{3}+\mathbf{5}+\mathbf{6}+\mathbf{7}+\mathbf{8} & \mathbf{2}+\mathbf{3}+\mathbf{4}+\mathbf{6}+\mathbf{7}+\mathbf{8} & \mathbf{2}+\mathbf{3}+\mathbf{4}+\mathbf{5}+\mathbf{7}+\mathbf{8} & \mathbf{2}+\mathbf{3}+\mathbf{4}+\mathbf{5}+\mathbf{6}+\mathbf{8} & \mathbf{1}+\mathbf{2}+\mathbf{3}+\mathbf{4}+\mathbf{5}+\mathbf{6}+\mathbf{7}+\mathbf{8} \\ \hline \end{array}\]The fusion rules are invariant under the group generated by the following permutations:
\[\{(\mathbf{2} \ \mathbf{3} \ \mathbf{5} \ \mathbf{6} \ \mathbf{4} \ \mathbf{7}), (\mathbf{2} \ \mathbf{7} \ \mathbf{4} \ \mathbf{6} \ \mathbf{5} \ \mathbf{3})\}\]Quantum Dimensions
| Particle | Numeric | Symbolic |
|---|---|---|
| \(\mathbf{1}\) | \(1.\) | \(1\) |
| \(\mathbf{2}\) | \(5.19258\) | \(\frac{1}{2} \left(5+\sqrt{29}\right)\) |
| \(\mathbf{3}\) | \(5.19258\) | \(\frac{1}{2} \left(5+\sqrt{29}\right)\) |
| \(\mathbf{4}\) | \(5.19258\) | \(\frac{1}{2} \left(5+\sqrt{29}\right)\) |
| \(\mathbf{5}\) | \(5.19258\) | \(\frac{1}{2} \left(5+\sqrt{29}\right)\) |
| \(\mathbf{6}\) | \(5.19258\) | \(\frac{1}{2} \left(5+\sqrt{29}\right)\) |
| \(\mathbf{7}\) | \(5.19258\) | \(\frac{1}{2} \left(5+\sqrt{29}\right)\) |
| \(\mathbf{8}\) | \(6.19258\) | \(\frac{1}{2} \left(7+\sqrt{29}\right)\) |
| \(\mathcal{D}_{FP}^2\) | \(201.126\) | \(1+\frac{3}{2} \left(5+\sqrt{29}\right)^2+\frac{1}{4} \left(7+\sqrt{29}\right)^2\) |
Characters
The symbolic character table is the following
\[\begin{array}{|cccccccc|} \hline \mathbf{1} & \mathbf{2} & \mathbf{3} & \mathbf{4} & \mathbf{5} & \mathbf{6} & \mathbf{7} & \mathbf{8} \\ \hline 1 & \frac{1}{2} \left(5+\sqrt{29}\right) & \frac{1}{2} \left(5+\sqrt{29}\right) & \frac{1}{2} \left(5+\sqrt{29}\right) & \frac{1}{2} \left(5+\sqrt{29}\right) & \frac{1}{2} \left(5+\sqrt{29}\right) & \frac{1}{2} \left(5+\sqrt{29}\right) & \frac{1}{2} \left(7+\sqrt{29}\right) \\ 1 & \text{Root}\left[x^6-x^5-5 x^4+4 x^3+6 x^2-3 x-1,2\right] & \text{Root}\left[x^6-x^5-5 x^4+4 x^3+6 x^2-3 x-1,4\right] & \text{Root}\left[x^6-x^5-5 x^4+4 x^3+6 x^2-3 x-1,6\right] & \text{Root}\left[x^6-x^5-5 x^4+4 x^3+6 x^2-3 x-1,5\right] & \text{Root}\left[x^6-x^5-5 x^4+4 x^3+6 x^2-3 x-1,3\right] & \text{Root}\left[x^6-x^5-5 x^4+4 x^3+6 x^2-3 x-1,1\right] & -1 \\ 1 & \text{Root}\left[x^6-x^5-5 x^4+4 x^3+6 x^2-3 x-1,6\right] & \text{Root}\left[x^6-x^5-5 x^4+4 x^3+6 x^2-3 x-1,1\right] & \text{Root}\left[x^6-x^5-5 x^4+4 x^3+6 x^2-3 x-1,5\right] & \text{Root}\left[x^6-x^5-5 x^4+4 x^3+6 x^2-3 x-1,2\right] & \text{Root}\left[x^6-x^5-5 x^4+4 x^3+6 x^2-3 x-1,4\right] & \text{Root}\left[x^6-x^5-5 x^4+4 x^3+6 x^2-3 x-1,3\right] & -1 \\ 1 & \text{Root}\left[x^6-x^5-5 x^4+4 x^3+6 x^2-3 x-1,1\right] & \text{Root}\left[x^6-x^5-5 x^4+4 x^3+6 x^2-3 x-1,2\right] & \text{Root}\left[x^6-x^5-5 x^4+4 x^3+6 x^2-3 x-1,3\right] & \text{Root}\left[x^6-x^5-5 x^4+4 x^3+6 x^2-3 x-1,4\right] & \text{Root}\left[x^6-x^5-5 x^4+4 x^3+6 x^2-3 x-1,5\right] & \text{Root}\left[x^6-x^5-5 x^4+4 x^3+6 x^2-3 x-1,6\right] & -1 \\ 1 & \text{Root}\left[x^6-x^5-5 x^4+4 x^3+6 x^2-3 x-1,4\right] & \text{Root}\left[x^6-x^5-5 x^4+4 x^3+6 x^2-3 x-1,5\right] & \text{Root}\left[x^6-x^5-5 x^4+4 x^3+6 x^2-3 x-1,1\right] & \text{Root}\left[x^6-x^5-5 x^4+4 x^3+6 x^2-3 x-1,3\right] & \text{Root}\left[x^6-x^5-5 x^4+4 x^3+6 x^2-3 x-1,6\right] & \text{Root}\left[x^6-x^5-5 x^4+4 x^3+6 x^2-3 x-1,2\right] & -1 \\ 1 & \text{Root}\left[x^6-x^5-5 x^4+4 x^3+6 x^2-3 x-1,5\right] & \text{Root}\left[x^6-x^5-5 x^4+4 x^3+6 x^2-3 x-1,3\right] & \text{Root}\left[x^6-x^5-5 x^4+4 x^3+6 x^2-3 x-1,2\right] & \text{Root}\left[x^6-x^5-5 x^4+4 x^3+6 x^2-3 x-1,6\right] & \text{Root}\left[x^6-x^5-5 x^4+4 x^3+6 x^2-3 x-1,1\right] & \text{Root}\left[x^6-x^5-5 x^4+4 x^3+6 x^2-3 x-1,4\right] & -1 \\ 1 & \text{Root}\left[x^6-x^5-5 x^4+4 x^3+6 x^2-3 x-1,3\right] & \text{Root}\left[x^6-x^5-5 x^4+4 x^3+6 x^2-3 x-1,6\right] & \text{Root}\left[x^6-x^5-5 x^4+4 x^3+6 x^2-3 x-1,4\right] & \text{Root}\left[x^6-x^5-5 x^4+4 x^3+6 x^2-3 x-1,1\right] & \text{Root}\left[x^6-x^5-5 x^4+4 x^3+6 x^2-3 x-1,2\right] & \text{Root}\left[x^6-x^5-5 x^4+4 x^3+6 x^2-3 x-1,5\right] & -1 \\ 1 & \frac{1}{2} \left(5-\sqrt{29}\right) & \frac{1}{2} \left(5-\sqrt{29}\right) & \frac{1}{2} \left(5-\sqrt{29}\right) & \frac{1}{2} \left(5-\sqrt{29}\right) & \frac{1}{2} \left(5-\sqrt{29}\right) & \frac{1}{2} \left(5-\sqrt{29}\right) & \frac{1}{2} \left(7-\sqrt{29}\right) \\ \hline \end{array}\]The numeric character table is the following
\[\begin{array}{|rrrrrrrr|} \hline \mathbf{1} & \mathbf{2} & \mathbf{3} & \mathbf{4} & \mathbf{5} & \mathbf{6} & \mathbf{7} & \mathbf{8} \\ \hline 1.000 & 5.193 & 5.193 & 5.193 & 5.193 & 5.193 & 5.193 & 6.193 \\ 1.000 & -1.136 & 0.7092 & 1.942 & 1.497 & -0.2411 & -1.771 & -1.000 \\ 1.000 & 1.942 & -1.771 & 1.497 & -1.136 & 0.7092 & -0.2411 & -1.000 \\ 1.000 & -1.771 & -1.136 & -0.2411 & 0.7092 & 1.497 & 1.942 & -1.000 \\ 1.000 & 0.7092 & 1.497 & -1.771 & -0.2411 & 1.942 & -1.136 & -1.000 \\ 1.000 & 1.497 & -0.2411 & -1.136 & 1.942 & -1.771 & 0.7092 & -1.000 \\ 1.000 & -0.2411 & 1.942 & 0.7092 & -1.771 & -1.136 & 1.497 & -1.000 \\ 1.000 & -0.1926 & -0.1926 & -0.1926 & -0.1926 & -0.1926 & -0.1926 & 0.8074 \\ \hline \end{array}\]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 $d$-number criterion.
Data
Download links for numeric data: