\(\text{FR}^{9,2}_{41}\)
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{6} & \mathbf{8} & \mathbf{7} & \mathbf{3} & \mathbf{5} & \mathbf{4} & \mathbf{9} \\ \mathbf{3} & \mathbf{6} & \mathbf{1}+\mathbf{4}+\mathbf{5}+\mathbf{7}+\mathbf{8} & \mathbf{3}+\mathbf{6}+\mathbf{8}+\mathbf{9} & \mathbf{3}+\mathbf{6}+\mathbf{7}+\mathbf{9} & \mathbf{2}+\mathbf{4}+\mathbf{5}+\mathbf{7}+\mathbf{8} & \mathbf{3}+\mathbf{5}+\mathbf{6}+\mathbf{9} & \mathbf{3}+\mathbf{4}+\mathbf{6}+\mathbf{9} & \mathbf{4}+\mathbf{5}+\mathbf{7}+\mathbf{8}+\mathbf{9} \\ \mathbf{4} & \mathbf{7} & \mathbf{3}+\mathbf{6}+\mathbf{7}+\mathbf{9} & \mathbf{1}+\mathbf{5}+\mathbf{6}+\mathbf{7}+\mathbf{8} & \mathbf{4}+\mathbf{5}+\mathbf{7}+\mathbf{9} & \mathbf{3}+\mathbf{4}+\mathbf{6}+\mathbf{9} & \mathbf{2}+\mathbf{3}+\mathbf{4}+\mathbf{5}+\mathbf{8} & \mathbf{4}+\mathbf{7}+\mathbf{8}+\mathbf{9} & \mathbf{3}+\mathbf{5}+\mathbf{6}+\mathbf{8}+\mathbf{9} \\ \mathbf{5} & \mathbf{8} & \mathbf{3}+\mathbf{6}+\mathbf{8}+\mathbf{9} & \mathbf{4}+\mathbf{5}+\mathbf{8}+\mathbf{9} & \mathbf{1}+\mathbf{4}+\mathbf{6}+\mathbf{7}+\mathbf{8} & \mathbf{3}+\mathbf{5}+\mathbf{6}+\mathbf{9} & \mathbf{5}+\mathbf{7}+\mathbf{8}+\mathbf{9} & \mathbf{2}+\mathbf{3}+\mathbf{4}+\mathbf{5}+\mathbf{7} & \mathbf{3}+\mathbf{4}+\mathbf{6}+\mathbf{7}+\mathbf{9} \\ \mathbf{6} & \mathbf{3} & \mathbf{2}+\mathbf{4}+\mathbf{5}+\mathbf{7}+\mathbf{8} & \mathbf{3}+\mathbf{4}+\mathbf{6}+\mathbf{9} & \mathbf{3}+\mathbf{5}+\mathbf{6}+\mathbf{9} & \mathbf{1}+\mathbf{4}+\mathbf{5}+\mathbf{7}+\mathbf{8} & \mathbf{3}+\mathbf{6}+\mathbf{7}+\mathbf{9} & \mathbf{3}+\mathbf{6}+\mathbf{8}+\mathbf{9} & \mathbf{4}+\mathbf{5}+\mathbf{7}+\mathbf{8}+\mathbf{9} \\ \mathbf{7} & \mathbf{4} & \mathbf{3}+\mathbf{4}+\mathbf{6}+\mathbf{9} & \mathbf{4}+\mathbf{7}+\mathbf{8}+\mathbf{9} & \mathbf{2}+\mathbf{3}+\mathbf{4}+\mathbf{5}+\mathbf{8} & \mathbf{3}+\mathbf{6}+\mathbf{7}+\mathbf{9} & \mathbf{4}+\mathbf{5}+\mathbf{7}+\mathbf{9} & \mathbf{1}+\mathbf{5}+\mathbf{6}+\mathbf{7}+\mathbf{8} & \mathbf{3}+\mathbf{5}+\mathbf{6}+\mathbf{8}+\mathbf{9} \\ \mathbf{8} & \mathbf{5} & \mathbf{3}+\mathbf{5}+\mathbf{6}+\mathbf{9} & \mathbf{2}+\mathbf{3}+\mathbf{4}+\mathbf{5}+\mathbf{7} & \mathbf{5}+\mathbf{7}+\mathbf{8}+\mathbf{9} & \mathbf{3}+\mathbf{6}+\mathbf{8}+\mathbf{9} & \mathbf{1}+\mathbf{4}+\mathbf{6}+\mathbf{7}+\mathbf{8} & \mathbf{4}+\mathbf{5}+\mathbf{8}+\mathbf{9} & \mathbf{3}+\mathbf{4}+\mathbf{6}+\mathbf{7}+\mathbf{9} \\ \mathbf{9} & \mathbf{9} & \mathbf{4}+\mathbf{5}+\mathbf{7}+\mathbf{8}+\mathbf{9} & \mathbf{3}+\mathbf{5}+\mathbf{6}+\mathbf{7}+\mathbf{9} & \mathbf{3}+\mathbf{4}+\mathbf{6}+\mathbf{8}+\mathbf{9} & \mathbf{4}+\mathbf{5}+\mathbf{7}+\mathbf{8}+\mathbf{9} & \mathbf{3}+\mathbf{4}+\mathbf{6}+\mathbf{8}+\mathbf{9} & \mathbf{3}+\mathbf{5}+\mathbf{6}+\mathbf{7}+\mathbf{9} & \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:
\[\{(\mathbf{4} \ \mathbf{5}) (\mathbf{7} \ \mathbf{8})\}\]The following elements form non-trivial sub fusion rings
| Elements | SubRing |
|---|---|
| \(\{\mathbf{1},\mathbf{2}\}\) | \(\mathbb{Z}_2:\ \text{FR}^{2,0}_{1}\) |
Frobenius-Perron Dimensions
| Particle | Numeric | Symbolic |
|---|---|---|
| \(\mathbf{1}\) | \(1.\) | \(1\) |
| \(\mathbf{2}\) | \(1.\) | \(1\) |
| \(\mathbf{3}\) | \(4.23607\) | \(2+\sqrt{5}\) |
| \(\mathbf{4}\) | \(4.23607\) | \(2+\sqrt{5}\) |
| \(\mathbf{5}\) | \(4.23607\) | \(2+\sqrt{5}\) |
| \(\mathbf{6}\) | \(4.23607\) | \(2+\sqrt{5}\) |
| \(\mathbf{7}\) | \(4.23607\) | \(2+\sqrt{5}\) |
| \(\mathbf{8}\) | \(4.23607\) | \(2+\sqrt{5}\) |
| \(\mathbf{9}\) | \(5.23607\) | \(3+\sqrt{5}\) |
| \(\mathcal{D}_{FP}^2\) | \(137.082\) | \(2+6 \left(2+\sqrt{5}\right)^2+\left(3+\sqrt{5}\right)^2\) |
Characters
The symbolic character table is the following
\[\begin{array}{|ccccccccc|} \hline \mathbf{1} & \mathbf{2} & \mathbf{4} & \mathbf{5} & \mathbf{3} & \mathbf{7} & \mathbf{8} & \mathbf{6} & \mathbf{9} \\ \hline 1 & 1 & 2+\sqrt{5} & 2+\sqrt{5} & 2+\sqrt{5} & 2+\sqrt{5} & 2+\sqrt{5} & 2+\sqrt{5} & 3+\sqrt{5} \\ 1 & 1 & 2 & 2 & -3 & 2 & 2 & -3 & -2 \\ 1 & 1 & 2-\sqrt{5} & 2-\sqrt{5} & 2-\sqrt{5} & 2-\sqrt{5} & 2-\sqrt{5} & 2-\sqrt{5} & 3-\sqrt{5} \\ 1 & -1 & 0 & 0 & 1 & 0 & 0 & -1 & 0 \\ 1 & -1 & 1 & 1 & -1 & -1 & -1 & 1 & 0 \\ 2 & -\frac{1}{6} & -\frac{7}{4} & -\frac{5}{12} & -\frac{1}{6} & \frac{13}{12} & \frac{13}{12} & 2 & -\frac{11}{6} \\ \hline \end{array}\]The numeric character table is the following
\[\begin{array}{|rrrrrrrrr|} \hline \mathbf{1} & \mathbf{2} & \mathbf{4} & \mathbf{5} & \mathbf{3} & \mathbf{7} & \mathbf{8} & \mathbf{6} & \mathbf{9} \\ \hline 1.000 & 1.000 & 4.236 & 4.236 & 4.236 & 4.236 & 4.236 & 4.236 & 5.236 \\ 1.000 & 1.000 & 2.000 & 2.000 & -3.000 & 2.000 & 2.000 & -3.000 & -2.000 \\ 1.000 & 1.000 & -0.2361 & -0.2361 & -0.2361 & -0.2361 & -0.2361 & -0.2361 & 0.7639 \\ 1.000 & -1.000 & 0 & 0 & 1.000 & 0 & 0 & -1.000 & 0 \\ 1.000 & -1.000 & 1.000 & 1.000 & -1.000 & -1.000 & -1.000 & 1.000 & 0 \\ 2.000 & -0.1667 & -1.750 & -0.4167 & -0.1667 & 1.083 & 1.083 & 2.000 & -1.833 \\ \hline \end{array}\]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
Data
Download links for numeric data: