\(\text{TriCritIsing}:\ \text{FR}^{6,0}_{4}\)
Fusion Rules
\[\begin{array}{|llllll|} \hline \mathbf{1} & \mathbf{2} & \mathbf{3} & \mathbf{4} & \mathbf{5} & \mathbf{6} \\ \mathbf{2} & \mathbf{1} & \mathbf{3} & \mathbf{5} & \mathbf{4} & \mathbf{6} \\ \mathbf{3} & \mathbf{3} & \mathbf{1}+\mathbf{2} & \mathbf{6} & \mathbf{6} & \mathbf{4}+\mathbf{5} \\ \mathbf{4} & \mathbf{5} & \mathbf{6} & \mathbf{1}+\mathbf{5} & \mathbf{2}+\mathbf{4} & \mathbf{3}+\mathbf{6} \\ \mathbf{5} & \mathbf{4} & \mathbf{6} & \mathbf{2}+\mathbf{4} & \mathbf{1}+\mathbf{5} & \mathbf{3}+\mathbf{6} \\ \mathbf{6} & \mathbf{6} & \mathbf{4}+\mathbf{5} & \mathbf{3}+\mathbf{6} & \mathbf{3}+\mathbf{6} & \mathbf{1}+\mathbf{2}+\mathbf{4}+\mathbf{5} \\ \hline \end{array}\]The following particles form non-trivial sub fusion rings
| Particles | SubRing |
|---|---|
| \(\{\mathbf{1},\mathbf{2}\}\) | \(\mathbb{Z}_2:\ \text{FR}^{2,0}_{1}\) |
| \(\{\mathbf{1},\mathbf{5}\}\) | \(\text{Fib}:\ \text{FR}^{2,0}_{2}\) |
| \(\{\mathbf{1},\mathbf{2},\mathbf{3}\}\) | \(\text{Ising}:\ \text{FR}^{3,0}_{1}\) |
| \(\{\mathbf{1},\mathbf{2},\mathbf{4},\mathbf{5}\}\) | \(\text{SU(2})_3:\ \text{FR}^{4,0}_{2}\) |
Quantum Dimensions
| Particle | Numeric | Symbolic |
|---|---|---|
| \(\mathbf{1}\) | \(1.\) | \(1\) |
| \(\mathbf{2}\) | \(1.\) | \(1\) |
| \(\mathbf{3}\) | \(1.41421\) | \(\sqrt{2}\) |
| \(\mathbf{4}\) | \(1.61803\) | \(\frac{1}{2} \left(1+\sqrt{5}\right)\) |
| \(\mathbf{5}\) | \(1.61803\) | \(\frac{1}{2} \left(1+\sqrt{5}\right)\) |
| \(\mathbf{6}\) | \(2.28825\) | \(\sqrt{3+\sqrt{5}}\) |
| \(\mathcal{D}_{FP}^2\) | \(14.4721\) | \(10+2\sqrt{5}\) |
Characters
The symbolic character table is the following
\[\begin{array}{|cccccc|} \hline \mathbf{1} & \mathbf{2} & \mathbf{3} & \mathbf{5} & \mathbf{4} & \mathbf{6} \\ \hline 1 & 1 & \sqrt{2} & \frac{1}{2} \left(1+\sqrt{5}\right) & \frac{1}{2} \left(1+\sqrt{5}\right) & \sqrt{3+\sqrt{5}} \\ 1 & 1 & -\sqrt{2} & \frac{1}{2} \left(1+\sqrt{5}\right) & \frac{1}{2} \left(1+\sqrt{5}\right) & -\sqrt{3+\sqrt{5}} \\ 1 & 1 & \sqrt{2} & \frac{1}{2} \left(1-\sqrt{5}\right) & \frac{1}{2} \left(1-\sqrt{5}\right) & -\sqrt{3-\sqrt{5}} \\ 1 & 1 & -\sqrt{2} & \frac{1}{2} \left(1-\sqrt{5}\right) & \frac{1}{2} \left(1-\sqrt{5}\right) & \sqrt{3-\sqrt{5}} \\ 1 & -1 & 0 & \frac{1}{2} \left(1+\sqrt{5}\right) & \frac{1}{2} \left(-1-\sqrt{5}\right) & 0 \\ 1 & -1 & 0 & \frac{1}{2} \left(1-\sqrt{5}\right) & \frac{1}{2} \left(\sqrt{5}-1\right) & 0 \\ \hline \end{array}\]The numeric character table is the following
\[\begin{array}{|rrrrrr|} \hline \mathbf{1} & \mathbf{2} & \mathbf{3} & \mathbf{5} & \mathbf{4} & \mathbf{6} \\ \hline 1.000 & 1.000 & 1.414 & 1.618 & 1.618 & 2.288 \\ 1.000 & 1.000 & -1.414 & 1.618 & 1.618 & -2.288 \\ 1.000 & 1.000 & 1.414 & -0.6180 & -0.6180 & -0.8740 \\ 1.000 & 1.000 & -1.414 & -0.6180 & -0.6180 & 0.8740 \\ 1.000 & -1.000 & 0 & 1.618 & -1.618 & 0 \\ 1.000 & -1.000 & 0 & -0.6180 & 0.6180 & 0 \\ \hline \end{array}\]Modular Data
The matching \(S\)-matrices and twist factors are the following
| \(S\)-matrix | Twist factors |
|---|---|
| \(\frac{1}{\sqrt{2 \left(5+\sqrt{5}\right)}}\left(\begin{array}{cccccc} 1 & 1 & \sqrt{2} & \frac{1}{2} \left(1+\sqrt{5}\right) & \frac{1}{2} \left(1+\sqrt{5}\right) & \sqrt{3+\sqrt{5}} \\ 1 & 1 & -\sqrt{2} & \frac{1}{2} \left(1+\sqrt{5}\right) & \frac{1}{2} \left(1+\sqrt{5}\right) & -\sqrt{3+\sqrt{5}} \\ \sqrt{2} & -\sqrt{2} & 0 & -\sqrt{3+\sqrt{5}} & \sqrt{3+\sqrt{5}} & 0 \\ \frac{1}{2} \left(1+\sqrt{5}\right) & \frac{1}{2} \left(1+\sqrt{5}\right) & -\sqrt{3+\sqrt{5}} & -1 & -1 & \sqrt{2} \\ \frac{1}{2} \left(1+\sqrt{5}\right) & \frac{1}{2} \left(1+\sqrt{5}\right) & \sqrt{3+\sqrt{5}} & -1 & -1 & -\sqrt{2} \\ \sqrt{3+\sqrt{5}} & -\sqrt{3+\sqrt{5}} & 0 & \sqrt{2} & -\sqrt{2} & 0 \\\end{array}\right)\) | \(\begin{array}{l}\left(0,\frac{1}{2},\frac{7}{16},\frac{1}{10},-\frac{2}{5},\frac{3}{80}\right) \\\left(0,\frac{1}{2},-\frac{5}{16},-\frac{1}{10},\frac{2}{5},\frac{7}{80}\right) \\\left(0,\frac{1}{2},-\frac{7}{16},\frac{1}{10},-\frac{2}{5},\frac{13}{80}\right) \\\left(0,\frac{1}{2},-\frac{3}{16},-\frac{1}{10},\frac{2}{5},\frac{17}{80}\right) \\\left(0,\frac{1}{2},-\frac{5}{16},\frac{1}{10},-\frac{2}{5},\frac{23}{80}\right) \\\left(0,\frac{1}{2},-\frac{1}{16},-\frac{1}{10},\frac{2}{5},\frac{27}{80}\right) \\\left(0,\frac{1}{2},-\frac{3}{16},\frac{1}{10},-\frac{2}{5},\frac{33}{80}\right) \\\left(0,\frac{1}{2},\frac{1}{16},-\frac{1}{10},\frac{2}{5},\frac{37}{80}\right) \\\left(0,\frac{1}{2},-\frac{1}{16},\frac{1}{10},-\frac{2}{5},-\frac{37}{80}\right) \\\left(0,\frac{1}{2},\frac{3}{16},-\frac{1}{10},\frac{2}{5},-\frac{33}{80}\right) \\\left(0,\frac{1}{2},\frac{1}{16},\frac{1}{10},-\frac{2}{5},-\frac{27}{80}\right) \\\left(0,\frac{1}{2},\frac{5}{16},-\frac{1}{10},\frac{2}{5},-\frac{23}{80}\right) \\\left(0,\frac{1}{2},\frac{3}{16},\frac{1}{10},-\frac{2}{5},-\frac{17}{80}\right) \\\left(0,\frac{1}{2},\frac{7}{16},-\frac{1}{10},\frac{2}{5},-\frac{13}{80}\right) \\\left(0,\frac{1}{2},\frac{5}{16},\frac{1}{10},-\frac{2}{5},-\frac{7}{80}\right) \\\left(0,\frac{1}{2},-\frac{7}{16},-\frac{1}{10},\frac{2}{5},-\frac{3}{80}\right)\end{array}\) |
Adjoint Subring
Particles \(\mathbf{1}, \mathbf{2}, \mathbf{4}, \mathbf{5}\), form the adjoint subring \(\text{SU(2})_3:\ \text{FR}^{4,0}_{2}\) .
The upper central series is the following: \(\text{TriCritIsing} \underset{ \mathbf{1}, \mathbf{2}, \mathbf{4}, \mathbf{5} }{\supset} \text{SU(2})_3 \underset{ \mathbf{1}, \mathbf{4} }{\supset} \text{Fib}\)
Universal grading
Each particle can be graded as follows: \(\text{deg}(\mathbf{1}) = \mathbf{1}', \text{deg}(\mathbf{2}) = \mathbf{1}', \text{deg}(\mathbf{3}) = \mathbf{2}', \text{deg}(\mathbf{4}) = \mathbf{1}', \text{deg}(\mathbf{5}) = \mathbf{1}', \text{deg}(\mathbf{6}) = \mathbf{2}'\), where the degrees form the group \(\mathbb{Z}_2\) with multiplication table:
\[\begin{array}{|ll|} \hline \mathbf{1}' & \mathbf{2}' \\ \mathbf{2}' & \mathbf{1}' \\ \hline \end{array}\]Categorifications
Data
Download links for numeric data: