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Q: FR18,6\text{Q}:\ \text{FR}^{8,6}_{1}

Fusion Rules

1234567821436587342178654312875656872134657812437856432187653412\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{4} & \mathbf{3} & \mathbf{6} & \mathbf{5} & \mathbf{8} & \mathbf{7} \\ \mathbf{3} & \mathbf{4} & \mathbf{2} & \mathbf{1} & \mathbf{7} & \mathbf{8} & \mathbf{6} & \mathbf{5} \\ \mathbf{4} & \mathbf{3} & \mathbf{1} & \mathbf{2} & \mathbf{8} & \mathbf{7} & \mathbf{5} & \mathbf{6} \\ \mathbf{5} & \mathbf{6} & \mathbf{8} & \mathbf{7} & \mathbf{2} & \mathbf{1} & \mathbf{3} & \mathbf{4} \\ \mathbf{6} & \mathbf{5} & \mathbf{7} & \mathbf{8} & \mathbf{1} & \mathbf{2} & \mathbf{4} & \mathbf{3} \\ \mathbf{7} & \mathbf{8} & \mathbf{5} & \mathbf{6} & \mathbf{4} & \mathbf{3} & \mathbf{2} & \mathbf{1} \\ \mathbf{8} & \mathbf{7} & \mathbf{6} & \mathbf{5} & \mathbf{3} & \mathbf{4} & \mathbf{1} & \mathbf{2} \\ \hline \end{array}

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

{(3 5 4 6),(3 6 4 5),(3 7 4 8),(3 8 4 7),(5 7 6 8),(5 8 6 7)}\{(\mathbf{3} \ \mathbf{5} \ \mathbf{4} \ \mathbf{6}), (\mathbf{3} \ \mathbf{6} \ \mathbf{4} \ \mathbf{5}), (\mathbf{3} \ \mathbf{7} \ \mathbf{4} \ \mathbf{8}), (\mathbf{3} \ \mathbf{8} \ \mathbf{4} \ \mathbf{7}), (\mathbf{5} \ \mathbf{7} \ \mathbf{6} \ \mathbf{8}), (\mathbf{5} \ \mathbf{8} \ \mathbf{6} \ \mathbf{7})\}

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}
{1,2,3,4}\{\mathbf{1},\mathbf{2},\mathbf{3},\mathbf{4}\} Z4: FR14,2\mathbb{Z}_4:\ \text{FR}^{4,2}_{1}
{1,2,5,6}\{\mathbf{1},\mathbf{2},\mathbf{5},\mathbf{6}\} Z4: FR14,2\mathbb{Z}_4:\ \text{FR}^{4,2}_{1}
{1,2,7,8}\{\mathbf{1},\mathbf{2},\mathbf{7},\mathbf{8}\} Z4: FR14,2\mathbb{Z}_4:\ \text{FR}^{4,2}_{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} 1.1. 11
8\mathbf{8} 1.1. 11
DFP2\mathcal{D}_{FP}^2 8.8. 88

Characters

The symbolic character table is the following

12345678111111111111−1−1−1−111−1−111−1−111−1−1−1−1112−2000000\begin{array}{|cccccccc|} \hline \mathbf{1} & \mathbf{2} & \mathbf{3} & \mathbf{4} & \mathbf{5} & \mathbf{6} & \mathbf{7} & \mathbf{8} \\ \hline 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\ 1 & 1 & 1 & 1 & -1 & -1 & -1 & -1 \\ 1 & 1 & -1 & -1 & 1 & 1 & -1 & -1 \\ 1 & 1 & -1 & -1 & -1 & -1 & 1 & 1 \\ 2 & -2 & 0 & 0 & 0 & 0 & 0 & 0 \\ \hline \end{array}

The numeric character table is the following

123456781.0001.0001.0001.0001.0001.0001.0001.0001.0001.0001.0001.000−1.000−1.000−1.000−1.0001.0001.000−1.000−1.0001.0001.000−1.000−1.0001.0001.000−1.000−1.000−1.000−1.0001.0001.0002.000−2.000000000\begin{array}{|rrrrrrrr|} \hline \mathbf{1} & \mathbf{2} & \mathbf{3} & \mathbf{4} & \mathbf{5} & \mathbf{6} & \mathbf{7} & \mathbf{8} \\ \hline 1.000 & 1.000 & 1.000 & 1.000 & 1.000 & 1.000 & 1.000 & 1.000 \\ 1.000 & 1.000 & 1.000 & 1.000 & -1.000 & -1.000 & -1.000 & -1.000 \\ 1.000 & 1.000 & -1.000 & -1.000 & 1.000 & 1.000 & -1.000 & -1.000 \\ 1.000 & 1.000 & -1.000 & -1.000 & -1.000 & -1.000 & 1.000 & 1.000 \\ 2.000 & -2.000 & 0 & 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

The adjoint subring is the trivial ring.

The upper central series is the following: Q⊃1Trivial\text{Q} \underset{ \mathbf{1} }{\supset} \text{Trivial}

Universal grading

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

1′2′3′4′5′6′7′8′2′1′4′3′6′5′8′7′3′4′2′1′7′8′6′5′4′3′1′2′8′7′5′6′5′6′8′7′2′1′3′4′6′5′7′8′1′2′4′3′7′8′5′6′4′3′2′1′8′7′6′5′3′4′1′2′\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{4}' & \mathbf{3}' & \mathbf{6}' & \mathbf{5}' & \mathbf{8}' & \mathbf{7}' \\ \mathbf{3}' & \mathbf{4}' & \mathbf{2}' & \mathbf{1}' & \mathbf{7}' & \mathbf{8}' & \mathbf{6}' & \mathbf{5}' \\ \mathbf{4}' & \mathbf{3}' & \mathbf{1}' & \mathbf{2}' & \mathbf{8}' & \mathbf{7}' & \mathbf{5}' & \mathbf{6}' \\ \mathbf{5}' & \mathbf{6}' & \mathbf{8}' & \mathbf{7}' & \mathbf{2}' & \mathbf{1}' & \mathbf{3}' & \mathbf{4}' \\ \mathbf{6}' & \mathbf{5}' & \mathbf{7}' & \mathbf{8}' & \mathbf{1}' & \mathbf{2}' & \mathbf{4}' & \mathbf{3}' \\ \mathbf{7}' & \mathbf{8}' & \mathbf{5}' & \mathbf{6}' & \mathbf{4}' & \mathbf{3}' & \mathbf{2}' & \mathbf{1}' \\ \mathbf{8}' & \mathbf{7}' & \mathbf{6}' & \mathbf{5}' & \mathbf{3}' & \mathbf{4}' & \mathbf{1}' & \mathbf{2}' \\ \hline \end{array}

Categorifications

Data

Download links for numeric data: