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FR178,2\text{FR}^{8,2}_{17}

Fusion Rules

1234567821345687331+2+35+64+64+57+87+8445+61+2+63+53+47+87+8554+63+51+2+43+67+87+8664+53+43+61+2+57+87+8787+87+87+87+82+3+4+5+6+7+81+3+4+5+6+7+8877+87+87+87+81+3+4+5+6+7+82+3+4+5+6+7+8\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{3} & \mathbf{4} & \mathbf{5} & \mathbf{6} & \mathbf{8} & \mathbf{7} \\ \mathbf{3} & \mathbf{3} & \mathbf{1}+\mathbf{2}+\mathbf{3} & \mathbf{5}+\mathbf{6} & \mathbf{4}+\mathbf{6} & \mathbf{4}+\mathbf{5} & \mathbf{7}+\mathbf{8} & \mathbf{7}+\mathbf{8} \\ \mathbf{4} & \mathbf{4} & \mathbf{5}+\mathbf{6} & \mathbf{1}+\mathbf{2}+\mathbf{6} & \mathbf{3}+\mathbf{5} & \mathbf{3}+\mathbf{4} & \mathbf{7}+\mathbf{8} & \mathbf{7}+\mathbf{8} \\ \mathbf{5} & \mathbf{5} & \mathbf{4}+\mathbf{6} & \mathbf{3}+\mathbf{5} & \mathbf{1}+\mathbf{2}+\mathbf{4} & \mathbf{3}+\mathbf{6} & \mathbf{7}+\mathbf{8} & \mathbf{7}+\mathbf{8} \\ \mathbf{6} & \mathbf{6} & \mathbf{4}+\mathbf{5} & \mathbf{3}+\mathbf{4} & \mathbf{3}+\mathbf{6} & \mathbf{1}+\mathbf{2}+\mathbf{5} & \mathbf{7}+\mathbf{8} & \mathbf{7}+\mathbf{8} \\ \mathbf{7} & \mathbf{8} & \mathbf{7}+\mathbf{8} & \mathbf{7}+\mathbf{8} & \mathbf{7}+\mathbf{8} & \mathbf{7}+\mathbf{8} & \mathbf{2}+\mathbf{3}+\mathbf{4}+\mathbf{5}+\mathbf{6}+\mathbf{7}+\mathbf{8} & \mathbf{1}+\mathbf{3}+\mathbf{4}+\mathbf{5}+\mathbf{6}+\mathbf{7}+\mathbf{8} \\ \mathbf{8} & \mathbf{7} & \mathbf{7}+\mathbf{8} & \mathbf{7}+\mathbf{8} & \mathbf{7}+\mathbf{8} & \mathbf{7}+\mathbf{8} & \mathbf{1}+\mathbf{3}+\mathbf{4}+\mathbf{5}+\mathbf{6}+\mathbf{7}+\mathbf{8} & \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:

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

The following particles form non-trivial sub fusion rings

Particles SubRing
{1,2}\{\mathbf{1},\mathbf{2}\} Z2: FR12,0\mathbb{Z}_2:\ \text{FR}^{2,0}_{1}
{1,2,3}\{\mathbf{1},\mathbf{2},\mathbf{3}\} Rep(D3): FR23,0\left.\text{Rep(}D_3\right):\ \text{FR}^{3,0}_{2}
{1,2,3,4,5,6}\{\mathbf{1},\mathbf{2},\mathbf{3},\mathbf{4},\mathbf{5},\mathbf{6}\} FR76,0\text{FR}^{6,0}_{7}

Quantum Dimensions

Particle Numeric Symbolic
1\mathbf{1} 1.1. 11
2\mathbf{2} 1.1. 11
3\mathbf{3} 2.2. 22
4\mathbf{4} 2.2. 22
5\mathbf{5} 2.2. 22
6\mathbf{6} 2.2. 22
7\mathbf{7} 4.162284.16228 1+101+\sqrt{10}
8\mathbf{8} 4.162284.16228 1+101+\sqrt{10}
DFP2\mathcal{D}_{FP}^2 52.649152.6491 18+2(1+10)218+2 \left(1+\sqrt{10}\right)^2

Characters

The symbolic character table is the following

123456871122221+101+101122221−101−1011−1Root[x3−3x+1,1]Root[x3−3x+1,2]Root[x3−3x+1,3]00112−1−1−10011−1Root[x3−3x+1,2]Root[x3−3x+1,3]Root[x3−3x+1,1]0011−1Root[x3−3x+1,3]Root[x3−3x+1,1]Root[x3−3x+1,2]001−10000i−i1−10000−ii\begin{array}{|cccccccc|} \hline \mathbf{1} & \mathbf{2} & \mathbf{3} & \mathbf{4} & \mathbf{5} & \mathbf{6} & \mathbf{8} & \mathbf{7} \\ \hline 1 & 1 & 2 & 2 & 2 & 2 & 1+\sqrt{10} & 1+\sqrt{10} \\ 1 & 1 & 2 & 2 & 2 & 2 & 1-\sqrt{10} & 1-\sqrt{10} \\ 1 & 1 & -1 & \text{Root}\left[x^3-3 x+1,1\right] & \text{Root}\left[x^3-3 x+1,2\right] & \text{Root}\left[x^3-3 x+1,3\right] & 0 & 0 \\ 1 & 1 & 2 & -1 & -1 & -1 & 0 & 0 \\ 1 & 1 & -1 & \text{Root}\left[x^3-3 x+1,2\right] & \text{Root}\left[x^3-3 x+1,3\right] & \text{Root}\left[x^3-3 x+1,1\right] & 0 & 0 \\ 1 & 1 & -1 & \text{Root}\left[x^3-3 x+1,3\right] & \text{Root}\left[x^3-3 x+1,1\right] & \text{Root}\left[x^3-3 x+1,2\right] & 0 & 0 \\ 1 & -1 & 0 & 0 & 0 & 0 & i & -i \\ 1 & -1 & 0 & 0 & 0 & 0 & -i & i \\ \hline \end{array}

The numeric character table is the following

123456871.0001.0002.0002.0002.0002.0004.1624.1621.0001.0002.0002.0002.0002.000−2.162−2.1621.0001.000−1.000−1.8790.34731.532001.0001.0002.000−1.000−1.000−1.000001.0001.000−1.0000.34731.532−1.879001.0001.000−1.0001.532−1.8790.3473001.000−1.00000001.000i−1.000i1.000−1.0000000−1.000i1.000i\begin{array}{|rrrrrrrr|} \hline \mathbf{1} & \mathbf{2} & \mathbf{3} & \mathbf{4} & \mathbf{5} & \mathbf{6} & \mathbf{8} & \mathbf{7} \\ \hline 1.000 & 1.000 & 2.000 & 2.000 & 2.000 & 2.000 & 4.162 & 4.162 \\ 1.000 & 1.000 & 2.000 & 2.000 & 2.000 & 2.000 & -2.162 & -2.162 \\ 1.000 & 1.000 & -1.000 & -1.879 & 0.3473 & 1.532 & 0 & 0 \\ 1.000 & 1.000 & 2.000 & -1.000 & -1.000 & -1.000 & 0 & 0 \\ 1.000 & 1.000 & -1.000 & 0.3473 & 1.532 & -1.879 & 0 & 0 \\ 1.000 & 1.000 & -1.000 & 1.532 & -1.879 & 0.3473 & 0 & 0 \\ 1.000 & -1.000 & 0 & 0 & 0 & 0 & 1.000 i & -1.000 i \\ 1.000 & -1.000 & 0 & 0 & 0 & 0 & -1.000 i & 1.000 i \\ \hline \end{array}

Modular Data

This fusion ring does not have any matching SS-and TT-matrices.

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

This fusion ring has no categorifications because of the dd-number criterion.

Data

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