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FR38,2\text{FR}^{8,2}_{3}

Fusion Rules

1234567821654387351624874651328753426178642315787888771+5+6+72+3+4+88777882+3+4+81+5+6+7\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{6} & \mathbf{5} & \mathbf{4} & \mathbf{3} & \mathbf{8} & \mathbf{7} \\ \mathbf{3} & \mathbf{5} & \mathbf{1} & \mathbf{6} & \mathbf{2} & \mathbf{4} & \mathbf{8} & \mathbf{7} \\ \mathbf{4} & \mathbf{6} & \mathbf{5} & \mathbf{1} & \mathbf{3} & \mathbf{2} & \mathbf{8} & \mathbf{7} \\ \mathbf{5} & \mathbf{3} & \mathbf{4} & \mathbf{2} & \mathbf{6} & \mathbf{1} & \mathbf{7} & \mathbf{8} \\ \mathbf{6} & \mathbf{4} & \mathbf{2} & \mathbf{3} & \mathbf{1} & \mathbf{5} & \mathbf{7} & \mathbf{8} \\ \mathbf{7} & \mathbf{8} & \mathbf{8} & \mathbf{8} & \mathbf{7} & \mathbf{7} & \mathbf{1}+\mathbf{5}+\mathbf{6}+\mathbf{7} & \mathbf{2}+\mathbf{3}+\mathbf{4}+\mathbf{8} \\ \mathbf{8} & \mathbf{7} & \mathbf{7} & \mathbf{7} & \mathbf{8} & \mathbf{8} & \mathbf{2}+\mathbf{3}+\mathbf{4}+\mathbf{8} & \mathbf{1}+\mathbf{5}+\mathbf{6}+\mathbf{7} \\ \hline \end{array}

The fusion rules are invariant under the group generated by the following permutations:

{(2 3 4),(2 4 3),(2 3)(5 6)}\{(\mathbf{2} \ \mathbf{3} \ \mathbf{4}), (\mathbf{2} \ \mathbf{4} \ \mathbf{3}), (\mathbf{2} \ \mathbf{3}) (\mathbf{5} \ \mathbf{6})\}

The following elements form non-trivial sub fusion rings

Elements SubRing
{1,2}\{\mathbf{1},\mathbf{2}\} Z2: FR12,0\mathbb{Z}_2:\ \text{FR}^{2,0}_{1}
{1,3}\{\mathbf{1},\mathbf{3}\} Z2: FR12,0\mathbb{Z}_2:\ \text{FR}^{2,0}_{1}
{1,4}\{\mathbf{1},\mathbf{4}\} Z2: FR12,0\mathbb{Z}_2:\ \text{FR}^{2,0}_{1}
{1,5,6}\{\mathbf{1},\mathbf{5},\mathbf{6}\} Z3: FR13,2\mathbb{Z}_3:\ \text{FR}^{3,2}_{1}
{1,5,6,7}\{\mathbf{1},\mathbf{5},\mathbf{6},\mathbf{7}\} Fib(Z3): FR34,2\left.\text{Fib(}\mathbb{Z}_3\right):\ \text{FR}^{4,2}_{3}
{1,2,3,4,5,6}\{\mathbf{1},\mathbf{2},\mathbf{3},\mathbf{4},\mathbf{5},\mathbf{6}\} D3: FR16,2D_3:\ \text{FR}^{6,2}_{1}

Frobenius-Perron Dimensions

Particle Numeric Symbolic
1\mathbf{1} 1.1. 11
2\mathbf{2} 1.1. 11
3\mathbf{3} 1.1. 11
4\mathbf{4} 1.1. 11
5\mathbf{5} 1.1. 11
6\mathbf{6} 1.1. 11
7\mathbf{7} 2.302782.30278 12(1+13)\frac{1}{2} \left(1+\sqrt{13}\right)
8\mathbf{8} 2.302782.30278 12(1+13)\frac{1}{2} \left(1+\sqrt{13}\right)
DFP2\mathcal{D}_{FP}^2 16.605616.6056 6+12(1+13)26+\frac{1}{2} \left(1+\sqrt{13}\right)^2

Characters

The symbolic character table is the following

1562347811111112(1+13)12(1+13)11111112(1−13)12(1−13)111−1−1−112(1+13)12(−1−13)111−1−1−112(1−13)12(13−1)2−1−100000\begin{array}{|cccccccc|} \hline \mathbf{1} & \mathbf{5} & \mathbf{6} & \mathbf{2} & \mathbf{3} & \mathbf{4} & \mathbf{7} & \mathbf{8} \\ \hline 1 & 1 & 1 & 1 & 1 & 1 & \frac{1}{2} \left(1+\sqrt{13}\right) & \frac{1}{2} \left(1+\sqrt{13}\right) \\ 1 & 1 & 1 & 1 & 1 & 1 & \frac{1}{2} \left(1-\sqrt{13}\right) & \frac{1}{2} \left(1-\sqrt{13}\right) \\ 1 & 1 & 1 & -1 & -1 & -1 & \frac{1}{2} \left(1+\sqrt{13}\right) & \frac{1}{2} \left(-1-\sqrt{13}\right) \\ 1 & 1 & 1 & -1 & -1 & -1 & \frac{1}{2} \left(1-\sqrt{13}\right) & \frac{1}{2} \left(\sqrt{13}-1\right) \\ 2 & -1 & -1 & 0 & 0 & 0 & 0 & 0 \\ \hline \end{array}

The numeric character table is the following

156234781.0001.0001.0001.0001.0001.0002.3032.3031.0001.0001.0001.0001.0001.000−1.303−1.3031.0001.0001.000−1.000−1.000−1.0002.303−2.3031.0001.0001.000−1.000−1.000−1.000−1.3031.3032.000−1.000−1.00000000\begin{array}{|rrrrrrrr|} \hline \mathbf{1} & \mathbf{5} & \mathbf{6} & \mathbf{2} & \mathbf{3} & \mathbf{4} & \mathbf{7} & \mathbf{8} \\ \hline 1.000 & 1.000 & 1.000 & 1.000 & 1.000 & 1.000 & 2.303 & 2.303 \\ 1.000 & 1.000 & 1.000 & 1.000 & 1.000 & 1.000 & -1.303 & -1.303 \\ 1.000 & 1.000 & 1.000 & -1.000 & -1.000 & -1.000 & 2.303 & -2.303 \\ 1.000 & 1.000 & 1.000 & -1.000 & -1.000 & -1.000 & -1.303 & 1.303 \\ 2.000 & -1.000 & -1.000 & 0 & 0 & 0 & 0 & 0 \\ \hline \end{array}

Representations of SL2(Z)SL_2(\mathbb{Z})

This fusion ring does not provide any representations of SL2(Z).SL_2(\mathbb{Z}).

Adjoint Subring

Elements 1,5,6,7\mathbf{1}, \mathbf{5}, \mathbf{6}, \mathbf{7}, form the adjoint subring Fib(Z3): FR34,2\left.\text{Fib(}\mathbb{Z}_3\right):\ \text{FR}^{4,2}_{3} .

The upper central series is the following: FR38,2⊃1,5,6,7Fib(Z3)\text{FR}^{8,2}_{3} \underset{ \mathbf{1}, \mathbf{5}, \mathbf{6}, \mathbf{7} }{\supset} \left.\text{Fib(}\mathbb{Z}_3\right)

Universal grading

Each particle can be graded as follows: deg(1)=1′,deg(2)=2′,deg(3)=2′,deg(4)=2′,deg(5)=1′,deg(6)=1′,deg(7)=1′,deg(8)=2′\text{deg}(\mathbf{1}) = \mathbf{1}', \text{deg}(\mathbf{2}) = \mathbf{2}', \text{deg}(\mathbf{3}) = \mathbf{2}', \text{deg}(\mathbf{4}) = \mathbf{2}', \text{deg}(\mathbf{5}) = \mathbf{1}', \text{deg}(\mathbf{6}) = \mathbf{1}', \text{deg}(\mathbf{7}) = \mathbf{1}', \text{deg}(\mathbf{8}) = \mathbf{2}', where the degrees form the group Z2\mathbb{Z}_2 with multiplication table:

1′2′2′1′\begin{array}{|ll|} \hline \mathbf{1}' & \mathbf{2}' \\ \mathbf{2}' & \mathbf{1}' \\ \hline \end{array}

Categorifications

Data

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