\(\text{Ising$\times $Ising}:\ \text{FR}^{9,0}_{3}\)
Fusion Rules
\[\begin{array}{|lllllllll|} \hline \mathbf{1} & \mathbf{2} & \mathbf{3} & \mathbf{4} & \mathbf{5} & \mathbf{6} & \mathbf{7} & \mathbf{8} & \mathbf{9} \\ \mathbf{2} & \mathbf{1} & \mathbf{4} & \mathbf{3} & \mathbf{8} & \mathbf{6} & \mathbf{7} & \mathbf{5} & \mathbf{9} \\ \mathbf{3} & \mathbf{4} & \mathbf{1} & \mathbf{2} & \mathbf{8} & \mathbf{7} & \mathbf{6} & \mathbf{5} & \mathbf{9} \\ \mathbf{4} & \mathbf{3} & \mathbf{2} & \mathbf{1} & \mathbf{5} & \mathbf{7} & \mathbf{6} & \mathbf{8} & \mathbf{9} \\ \mathbf{5} & \mathbf{8} & \mathbf{8} & \mathbf{5} & \mathbf{1}+\mathbf{4} & \mathbf{9} & \mathbf{9} & \mathbf{2}+\mathbf{3} & \mathbf{6}+\mathbf{7} \\ \mathbf{6} & \mathbf{6} & \mathbf{7} & \mathbf{7} & \mathbf{9} & \mathbf{1}+\mathbf{2} & \mathbf{3}+\mathbf{4} & \mathbf{9} & \mathbf{5}+\mathbf{8} \\ \mathbf{7} & \mathbf{7} & \mathbf{6} & \mathbf{6} & \mathbf{9} & \mathbf{3}+\mathbf{4} & \mathbf{1}+\mathbf{2} & \mathbf{9} & \mathbf{5}+\mathbf{8} \\ \mathbf{8} & \mathbf{5} & \mathbf{5} & \mathbf{8} & \mathbf{2}+\mathbf{3} & \mathbf{9} & \mathbf{9} & \mathbf{1}+\mathbf{4} & \mathbf{6}+\mathbf{7} \\ \mathbf{9} & \mathbf{9} & \mathbf{9} & \mathbf{9} & \mathbf{6}+\mathbf{7} & \mathbf{5}+\mathbf{8} & \mathbf{5}+\mathbf{8} & \mathbf{6}+\mathbf{7} & \mathbf{1}+\mathbf{2}+\mathbf{3}+\mathbf{4} \\ \hline \end{array}\]The fusion rules are invariant under the group generated by the following permutations:
\[\{(\mathbf{5} \ \mathbf{8}), (\mathbf{6} \ \mathbf{7}), (\mathbf{2} \ \mathbf{4}) (\mathbf{5} \ \mathbf{6} \ \mathbf{8} \ \mathbf{7}), (\mathbf{2} \ \mathbf{4}) (\mathbf{5} \ \mathbf{7} \ \mathbf{8} \ \mathbf{6})\}\]The following elements form non-trivial sub fusion rings
| Elements | SubRing |
|---|---|
| \(\{\mathbf{1},\mathbf{2}\}\) | \(\mathbb{Z}_2:\ \text{FR}^{2,0}_{1}\) |
| \(\{\mathbf{1},\mathbf{3}\}\) | \(\mathbb{Z}_2:\ \text{FR}^{2,0}_{1}\) |
| \(\{\mathbf{1},\mathbf{4}\}\) | \(\mathbb{Z}_2:\ \text{FR}^{2,0}_{1}\) |
| \(\{\mathbf{1},\mathbf{2},\mathbf{6}\}\) | \(\text{Ising}:\ \text{FR}^{3,0}_{1}\) |
| \(\{\mathbf{1},\mathbf{2},\mathbf{7}\}\) | \(\text{Ising}:\ \text{FR}^{3,0}_{1}\) |
| \(\{\mathbf{1},\mathbf{4},\mathbf{5}\}\) | \(\text{Ising}:\ \text{FR}^{3,0}_{1}\) |
| \(\{\mathbf{1},\mathbf{4},\mathbf{8}\}\) | \(\text{Ising}:\ \text{FR}^{3,0}_{1}\) |
| \(\{\mathbf{1},\mathbf{2},\mathbf{3},\mathbf{4}\}\) | \(\mathbb{Z}_2\times \mathbb{Z}_2:\ \text{FR}^{4,0}_{1}\) |
| \(\{\mathbf{1},\mathbf{2},\mathbf{3},\mathbf{4},\mathbf{9}\}\) | \(\left.\text{Rep(}D_4\right):\ \text{FR}^{5,0}_{1}\) |
| \(\{\mathbf{1},\mathbf{2},\mathbf{3},\mathbf{4},\mathbf{5},\mathbf{8}\}\) | \(\mathbb{Z}_2\text{$\times $Ising}:\ \text{FR}^{6,0}_{1}\) |
| \(\{\mathbf{1},\mathbf{2},\mathbf{3},\mathbf{4},\mathbf{6},\mathbf{7}\}\) | \(\mathbb{Z}_2\text{$\times $Ising}:\ \text{FR}^{6,0}_{1}\) |
Frobenius-Perron Dimensions
| Particle | Numeric | Symbolic |
|---|---|---|
| \(\mathbf{1}\) | \(1.\) | \(1\) |
| \(\mathbf{2}\) | \(1.\) | \(1\) |
| \(\mathbf{3}\) | \(1.\) | \(1\) |
| \(\mathbf{4}\) | \(1.\) | \(1\) |
| \(\mathbf{5}\) | \(1.41421\) | \(\sqrt{2}\) |
| \(\mathbf{6}\) | \(1.41421\) | \(\sqrt{2}\) |
| \(\mathbf{7}\) | \(1.41421\) | \(\sqrt{2}\) |
| \(\mathbf{8}\) | \(1.41421\) | \(\sqrt{2}\) |
| \(\mathbf{9}\) | \(2.\) | \(2\) |
| \(\mathcal{D}_{FP}^2\) | \(16.\) | \(16\) |
Characters
The symbolic character table is the following
\[\begin{array}{|ccccccccc|} \hline \mathbf{1} & \mathbf{2} & \mathbf{3} & \mathbf{4} & \mathbf{8} & \mathbf{6} & \mathbf{7} & \mathbf{5} & \mathbf{9} \\ \hline 1 & 1 & 1 & 1 & \sqrt{2} & \sqrt{2} & \sqrt{2} & \sqrt{2} & 2 \\ 1 & 1 & 1 & 1 & -\sqrt{2} & \sqrt{2} & \sqrt{2} & -\sqrt{2} & -2 \\ 1 & 1 & 1 & 1 & \sqrt{2} & -\sqrt{2} & -\sqrt{2} & \sqrt{2} & -2 \\ 1 & 1 & 1 & 1 & -\sqrt{2} & -\sqrt{2} & -\sqrt{2} & -\sqrt{2} & 2 \\ 1 & -1 & 1 & -1 & 0 & 0 & 0 & 0 & 0 \\ 1 & 1 & -1 & -1 & 0 & \sqrt{2} & -\sqrt{2} & 0 & 0 \\ 1 & -1 & -1 & 1 & \sqrt{2} & 0 & 0 & -\sqrt{2} & 0 \\ 1 & 1 & -1 & -1 & 0 & -\sqrt{2} & \sqrt{2} & 0 & 0 \\ 1 & -1 & -1 & 1 & -\sqrt{2} & 0 & 0 & \sqrt{2} & 0 \\ \hline \end{array}\]The numeric character table is the following
\[\begin{array}{|rrrrrrrrr|} \hline \mathbf{1} & \mathbf{2} & \mathbf{3} & \mathbf{4} & \mathbf{8} & \mathbf{6} & \mathbf{7} & \mathbf{5} & \mathbf{9} \\ \hline 1.000 & 1.000 & 1.000 & 1.000 & 1.414 & 1.414 & 1.414 & 1.414 & 2.000 \\ 1.000 & 1.000 & 1.000 & 1.000 & -1.414 & 1.414 & 1.414 & -1.414 & -2.000 \\ 1.000 & 1.000 & 1.000 & 1.000 & 1.414 & -1.414 & -1.414 & 1.414 & -2.000 \\ 1.000 & 1.000 & 1.000 & 1.000 & -1.414 & -1.414 & -1.414 & -1.414 & 2.000 \\ 1.000 & -1.000 & 1.000 & -1.000 & 0 & 0 & 0 & 0 & 0 \\ 1.000 & 1.000 & -1.000 & -1.000 & 0 & 1.414 & -1.414 & 0 & 0 \\ 1.000 & -1.000 & -1.000 & 1.000 & 1.414 & 0 & 0 & -1.414 & 0 \\ 1.000 & 1.000 & -1.000 & -1.000 & 0 & -1.414 & 1.414 & 0 & 0 \\ 1.000 & -1.000 & -1.000 & 1.000 & -1.414 & 0 & 0 & 1.414 & 0 \\ \hline \end{array}\]Representations of $SL_2(\mathbb{Z})$
The matching \(S\)-matrices and twist factors are the following
| \(S\)-matrix | Twist factors |
|---|---|
| \(\frac{1}{4}\left(\begin{array}{ccccccccc} 1 & 1 & 1 & 1 & \sqrt{2} & \sqrt{2} & \sqrt{2} & \sqrt{2} & 2 \\ 1 & 1 & 1 & 1 & \sqrt{2} & -\sqrt{2} & -\sqrt{2} & \sqrt{2} & -2 \\ 1 & 1 & 1 & 1 & -\sqrt{2} & -\sqrt{2} & -\sqrt{2} & -\sqrt{2} & 2 \\ 1 & 1 & 1 & 1 & -\sqrt{2} & \sqrt{2} & \sqrt{2} & -\sqrt{2} & -2 \\ \sqrt{2} & \sqrt{2} & -\sqrt{2} & -\sqrt{2} & 0 & 2 & -2 & 0 & 0 \\ \sqrt{2} & -\sqrt{2} & -\sqrt{2} & \sqrt{2} & 2 & 0 & 0 & -2 & 0 \\ \sqrt{2} & -\sqrt{2} & -\sqrt{2} & \sqrt{2} & -2 & 0 & 0 & 2 & 0 \\ \sqrt{2} & \sqrt{2} & -\sqrt{2} & -\sqrt{2} & 0 & -2 & 2 & 0 & 0 \\ 2 & -2 & 2 & -2 & 0 & 0 & 0 & 0 & 0 \\\end{array}\right)\) | \(\begin{array}{l}\left(0,\frac{1}{2},0,\frac{1}{2},-\frac{7}{16},-\frac{7}{16},\frac{1}{16},\frac{1}{16},\frac{1}{8}\right) \\\left(0,\frac{1}{2},0,\frac{1}{2},\frac{1}{16},\frac{1}{16},-\frac{7}{16},-\frac{7}{16},\frac{1}{8}\right) \\\left(0,\frac{1}{2},0,\frac{1}{2},-\frac{5}{16},-\frac{7}{16},\frac{1}{16},\frac{3}{16},\frac{1}{4}\right) \\\left(0,\frac{1}{2},0,\frac{1}{2},\frac{3}{16},\frac{1}{16},-\frac{7}{16},-\frac{5}{16},\frac{1}{4}\right) \\\left(0,\frac{1}{2},0,\frac{1}{2},-\frac{7}{16},-\frac{5}{16},\frac{3}{16},\frac{1}{16},\frac{1}{4}\right) \\\left(0,\frac{1}{2},0,\frac{1}{2},\frac{1}{16},\frac{3}{16},-\frac{5}{16},-\frac{7}{16},\frac{1}{4}\right) \\\left(0,\frac{1}{2},0,\frac{1}{2},-\frac{3}{16},-\frac{7}{16},\frac{1}{16},\frac{5}{16},\frac{3}{8}\right) \\\left(0,\frac{1}{2},0,\frac{1}{2},\frac{5}{16},\frac{1}{16},-\frac{7}{16},-\frac{3}{16},\frac{3}{8}\right) \\\left(0,\frac{1}{2},0,\frac{1}{2},-\frac{7}{16},-\frac{3}{16},\frac{5}{16},\frac{1}{16},\frac{3}{8}\right) \\\left(0,\frac{1}{2},0,\frac{1}{2},\frac{1}{16},\frac{5}{16},-\frac{3}{16},-\frac{7}{16},\frac{3}{8}\right) \\\left(0,\frac{1}{2},0,\frac{1}{2},-\frac{1}{16},-\frac{7}{16},\frac{1}{16},\frac{7}{16},\frac{1}{2}\right) \\\left(0,\frac{1}{2},0,\frac{1}{2},\frac{7}{16},\frac{1}{16},-\frac{7}{16},-\frac{1}{16},\frac{1}{2}\right) \\\left(0,\frac{1}{2},0,\frac{1}{2},-\frac{7}{16},-\frac{1}{16},\frac{7}{16},\frac{1}{16},\frac{1}{2}\right) \\\left(0,\frac{1}{2},0,\frac{1}{2},\frac{1}{16},\frac{7}{16},-\frac{1}{16},-\frac{7}{16},\frac{1}{2}\right) \\\left(0,\frac{1}{2},0,\frac{1}{2},\frac{1}{16},-\frac{7}{16},\frac{1}{16},-\frac{7}{16},-\frac{3}{8}\right) \\\left(0,\frac{1}{2},0,\frac{1}{2},-\frac{7}{16},\frac{1}{16},-\frac{7}{16},\frac{1}{16},-\frac{3}{8}\right) \\\left(0,\frac{1}{2},0,\frac{1}{2},\frac{3}{16},-\frac{7}{16},\frac{1}{16},-\frac{5}{16},-\frac{1}{4}\right) \\\left(0,\frac{1}{2},0,\frac{1}{2},-\frac{5}{16},\frac{1}{16},-\frac{7}{16},\frac{3}{16},-\frac{1}{4}\right) \\\left(0,\frac{1}{2},0,\frac{1}{2},-\frac{7}{16},\frac{3}{16},-\frac{5}{16},\frac{1}{16},-\frac{1}{4}\right) \\\left(0,\frac{1}{2},0,\frac{1}{2},\frac{1}{16},-\frac{5}{16},\frac{3}{16},-\frac{7}{16},-\frac{1}{4}\right) \\\left(0,\frac{1}{2},0,\frac{1}{2},\frac{5}{16},-\frac{7}{16},\frac{1}{16},-\frac{3}{16},-\frac{1}{8}\right) \\\left(0,\frac{1}{2},0,\frac{1}{2},-\frac{3}{16},\frac{1}{16},-\frac{7}{16},\frac{5}{16},-\frac{1}{8}\right) \\\left(0,\frac{1}{2},0,\frac{1}{2},-\frac{7}{16},\frac{5}{16},-\frac{3}{16},\frac{1}{16},-\frac{1}{8}\right) \\\left(0,\frac{1}{2},0,\frac{1}{2},\frac{1}{16},-\frac{3}{16},\frac{5}{16},-\frac{7}{16},-\frac{1}{8}\right) \\\left(0,\frac{1}{2},0,\frac{1}{2},\frac{7}{16},-\frac{7}{16},\frac{1}{16},-\frac{1}{16},0\right) \\\left(0,\frac{1}{2},0,\frac{1}{2},-\frac{1}{16},\frac{1}{16},-\frac{7}{16},\frac{7}{16},0\right) \\\left(0,\frac{1}{2},0,\frac{1}{2},-\frac{7}{16},\frac{7}{16},-\frac{1}{16},\frac{1}{16},0\right) \\\left(0,\frac{1}{2},0,\frac{1}{2},\frac{1}{16},-\frac{1}{16},\frac{7}{16},-\frac{7}{16},0\right) \\\left(0,\frac{1}{2},0,\frac{1}{2},-\frac{5}{16},-\frac{5}{16},\frac{3}{16},\frac{3}{16},\frac{3}{8}\right) \\\left(0,\frac{1}{2},0,\frac{1}{2},\frac{3}{16},\frac{3}{16},-\frac{5}{16},-\frac{5}{16},\frac{3}{8}\right) \\\left(0,\frac{1}{2},0,\frac{1}{2},-\frac{3}{16},-\frac{5}{16},\frac{3}{16},\frac{5}{16},\frac{1}{2}\right) \\\left(0,\frac{1}{2},0,\frac{1}{2},\frac{5}{16},\frac{3}{16},-\frac{5}{16},-\frac{3}{16},\frac{1}{2}\right) \\\left(0,\frac{1}{2},0,\frac{1}{2},-\frac{5}{16},-\frac{3}{16},\frac{5}{16},\frac{3}{16},\frac{1}{2}\right) \\\left(0,\frac{1}{2},0,\frac{1}{2},\frac{3}{16},\frac{5}{16},-\frac{3}{16},-\frac{5}{16},\frac{1}{2}\right) \\\left(0,\frac{1}{2},0,\frac{1}{2},-\frac{1}{16},-\frac{5}{16},\frac{3}{16},\frac{7}{16},-\frac{3}{8}\right) \\\left(0,\frac{1}{2},0,\frac{1}{2},\frac{7}{16},\frac{3}{16},-\frac{5}{16},-\frac{1}{16},-\frac{3}{8}\right) \\\left(0,\frac{1}{2},0,\frac{1}{2},-\frac{5}{16},-\frac{1}{16},\frac{7}{16},\frac{3}{16},-\frac{3}{8}\right) \\\left(0,\frac{1}{2},0,\frac{1}{2},\frac{3}{16},\frac{7}{16},-\frac{1}{16},-\frac{5}{16},-\frac{3}{8}\right) \\\left(0,\frac{1}{2},0,\frac{1}{2},\frac{3}{16},-\frac{5}{16},\frac{3}{16},-\frac{5}{16},-\frac{1}{8}\right) \\\left(0,\frac{1}{2},0,\frac{1}{2},-\frac{5}{16},\frac{3}{16},-\frac{5}{16},\frac{3}{16},-\frac{1}{8}\right) \\\left(0,\frac{1}{2},0,\frac{1}{2},\frac{5}{16},-\frac{5}{16},\frac{3}{16},-\frac{3}{16},0\right) \\\left(0,\frac{1}{2},0,\frac{1}{2},-\frac{3}{16},\frac{3}{16},-\frac{5}{16},\frac{5}{16},0\right) \\\left(0,\frac{1}{2},0,\frac{1}{2},-\frac{5}{16},\frac{5}{16},-\frac{3}{16},\frac{3}{16},0\right) \\\left(0,\frac{1}{2},0,\frac{1}{2},\frac{3}{16},-\frac{3}{16},\frac{5}{16},-\frac{5}{16},0\right) \\\left(0,\frac{1}{2},0,\frac{1}{2},\frac{7}{16},-\frac{5}{16},\frac{3}{16},-\frac{1}{16},\frac{1}{8}\right) \\\left(0,\frac{1}{2},0,\frac{1}{2},-\frac{1}{16},\frac{3}{16},-\frac{5}{16},\frac{7}{16},\frac{1}{8}\right) \\\left(0,\frac{1}{2},0,\frac{1}{2},-\frac{5}{16},\frac{7}{16},-\frac{1}{16},\frac{3}{16},\frac{1}{8}\right) \\\left(0,\frac{1}{2},0,\frac{1}{2},\frac{3}{16},-\frac{1}{16},\frac{7}{16},-\frac{5}{16},\frac{1}{8}\right) \\\left(0,\frac{1}{2},0,\frac{1}{2},-\frac{3}{16},-\frac{3}{16},\frac{5}{16},\frac{5}{16},-\frac{3}{8}\right) \\\left(0,\frac{1}{2},0,\frac{1}{2},\frac{5}{16},\frac{5}{16},-\frac{3}{16},-\frac{3}{16},-\frac{3}{8}\right) \\\left(0,\frac{1}{2},0,\frac{1}{2},-\frac{1}{16},-\frac{3}{16},\frac{5}{16},\frac{7}{16},-\frac{1}{4}\right) \\\left(0,\frac{1}{2},0,\frac{1}{2},\frac{7}{16},\frac{5}{16},-\frac{3}{16},-\frac{1}{16},-\frac{1}{4}\right) \\\left(0,\frac{1}{2},0,\frac{1}{2},-\frac{3}{16},-\frac{1}{16},\frac{7}{16},\frac{5}{16},-\frac{1}{4}\right) \\\left(0,\frac{1}{2},0,\frac{1}{2},\frac{5}{16},\frac{7}{16},-\frac{1}{16},-\frac{3}{16},-\frac{1}{4}\right) \\\left(0,\frac{1}{2},0,\frac{1}{2},\frac{5}{16},-\frac{3}{16},\frac{5}{16},-\frac{3}{16},\frac{1}{8}\right) \\\left(0,\frac{1}{2},0,\frac{1}{2},-\frac{3}{16},\frac{5}{16},-\frac{3}{16},\frac{5}{16},\frac{1}{8}\right) \\\left(0,\frac{1}{2},0,\frac{1}{2},\frac{7}{16},-\frac{3}{16},\frac{5}{16},-\frac{1}{16},\frac{1}{4}\right) \\\left(0,\frac{1}{2},0,\frac{1}{2},-\frac{1}{16},\frac{5}{16},-\frac{3}{16},\frac{7}{16},\frac{1}{4}\right) \\\left(0,\frac{1}{2},0,\frac{1}{2},-\frac{3}{16},\frac{7}{16},-\frac{1}{16},\frac{5}{16},\frac{1}{4}\right) \\\left(0,\frac{1}{2},0,\frac{1}{2},\frac{5}{16},-\frac{1}{16},\frac{7}{16},-\frac{3}{16},\frac{1}{4}\right) \\\left(0,\frac{1}{2},0,\frac{1}{2},-\frac{1}{16},-\frac{1}{16},\frac{7}{16},\frac{7}{16},-\frac{1}{8}\right) \\\left(0,\frac{1}{2},0,\frac{1}{2},\frac{7}{16},\frac{7}{16},-\frac{1}{16},-\frac{1}{16},-\frac{1}{8}\right) \\\left(0,\frac{1}{2},0,\frac{1}{2},\frac{7}{16},-\frac{1}{16},\frac{7}{16},-\frac{1}{16},\frac{3}{8}\right) \\\left(0,\frac{1}{2},0,\frac{1}{2},-\frac{1}{16},\frac{7}{16},-\frac{1}{16},\frac{7}{16},\frac{3}{8}\right)\end{array}\) |
Adjoint Subring
Elements \(\mathbf{1}, \mathbf{2}, \mathbf{3}, \mathbf{4}\), form the adjoint subring \(\mathbb{Z}_2\times \mathbb{Z}_2:\ \text{FR}^{4,0}_{1}\) .
The upper central series is the following: \(\text{Ising$\times $Ising} \underset{ \mathbf{1}, \mathbf{2}, \mathbf{3}, \mathbf{4} }{\supset} \mathbb{Z}_2\times \mathbb{Z}_2 \underset{ \mathbf{1} }{\supset} \text{Trivial}\)
Universal grading
Each particle can be graded as follows: \(\text{deg}(\mathbf{1}) = \mathbf{1}', \text{deg}(\mathbf{2}) = \mathbf{1}', \text{deg}(\mathbf{3}) = \mathbf{1}', \text{deg}(\mathbf{4}) = \mathbf{1}', \text{deg}(\mathbf{5}) = \mathbf{2}', \text{deg}(\mathbf{6}) = \mathbf{3}', \text{deg}(\mathbf{7}) = \mathbf{3}', \text{deg}(\mathbf{8}) = \mathbf{2}', \text{deg}(\mathbf{9}) = \mathbf{4}'\), where the degrees form the group \(\mathbb{Z}_2\times \mathbb{Z}_2\) with multiplication table:
\[\begin{array}{|llll|} \hline \mathbf{1}' & \mathbf{2}' & \mathbf{3}' & \mathbf{4}' \\ \mathbf{2}' & \mathbf{1}' & \mathbf{4}' & \mathbf{3}' \\ \mathbf{3}' & \mathbf{4}' & \mathbf{1}' & \mathbf{2}' \\ \mathbf{4}' & \mathbf{3}' & \mathbf{2}' & \mathbf{1}' \\ \hline \end{array}\]Categorifications
Data
Download links for numeric data: