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FR168,2\text{FR}^{8,2}_{16}

Fusion Rules

1234567821436587341256874321657856561+3+5+7+82+4+6+7+85+6+7+85+6+7+865652+4+6+7+81+3+5+7+85+6+7+85+6+7+878875+6+7+85+6+7+82+3+5+6+7+81+4+5+6+7+887785+6+7+85+6+7+81+4+5+6+7+82+3+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{4} & \mathbf{3} & \mathbf{6} & \mathbf{5} & \mathbf{8} & \mathbf{7} \\ \mathbf{3} & \mathbf{4} & \mathbf{1} & \mathbf{2} & \mathbf{5} & \mathbf{6} & \mathbf{8} & \mathbf{7} \\ \mathbf{4} & \mathbf{3} & \mathbf{2} & \mathbf{1} & \mathbf{6} & \mathbf{5} & \mathbf{7} & \mathbf{8} \\ \mathbf{5} & \mathbf{6} & \mathbf{5} & \mathbf{6} & \mathbf{1}+\mathbf{3}+\mathbf{5}+\mathbf{7}+\mathbf{8} & \mathbf{2}+\mathbf{4}+\mathbf{6}+\mathbf{7}+\mathbf{8} & \mathbf{5}+\mathbf{6}+\mathbf{7}+\mathbf{8} & \mathbf{5}+\mathbf{6}+\mathbf{7}+\mathbf{8} \\ \mathbf{6} & \mathbf{5} & \mathbf{6} & \mathbf{5} & \mathbf{2}+\mathbf{4}+\mathbf{6}+\mathbf{7}+\mathbf{8} & \mathbf{1}+\mathbf{3}+\mathbf{5}+\mathbf{7}+\mathbf{8} & \mathbf{5}+\mathbf{6}+\mathbf{7}+\mathbf{8} & \mathbf{5}+\mathbf{6}+\mathbf{7}+\mathbf{8} \\ \mathbf{7} & \mathbf{8} & \mathbf{8} & \mathbf{7} & \mathbf{5}+\mathbf{6}+\mathbf{7}+\mathbf{8} & \mathbf{5}+\mathbf{6}+\mathbf{7}+\mathbf{8} & \mathbf{2}+\mathbf{3}+\mathbf{5}+\mathbf{6}+\mathbf{7}+\mathbf{8} & \mathbf{1}+\mathbf{4}+\mathbf{5}+\mathbf{6}+\mathbf{7}+\mathbf{8} \\ \mathbf{8} & \mathbf{7} & \mathbf{7} & \mathbf{8} & \mathbf{5}+\mathbf{6}+\mathbf{7}+\mathbf{8} & \mathbf{5}+\mathbf{6}+\mathbf{7}+\mathbf{8} & \mathbf{1}+\mathbf{4}+\mathbf{5}+\mathbf{6}+\mathbf{7}+\mathbf{8} & \mathbf{2}+\mathbf{3}+\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)}\{(\mathbf{7} \ \mathbf{8})\}

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}
{1,2,3,4}\{\mathbf{1},\mathbf{2},\mathbf{3},\mathbf{4}\} Z2×Z2: FR14,0\mathbb{Z}_2\times \mathbb{Z}_2:\ \text{FR}^{4,0}_{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} 3.778463.77846 Root[x3−3x2−4x+4,3]\text{Root}\left[x^3-3 x^2-4 x+4,3\right]
6\mathbf{6} 3.778463.77846 Root[x3−3x2−4x+4,3]\text{Root}\left[x^3-3 x^2-4 x+4,3\right]
7\mathbf{7} 4.249144.24914 Root[x3−4x2−2x+4,3]\text{Root}\left[x^3-4 x^2-2 x+4,3\right]
8\mathbf{8} 4.249144.24914 Root[x3−4x2−2x+4,3]\text{Root}\left[x^3-4 x^2-2 x+4,3\right]
DFP2\mathcal{D}_{FP}^2 68.663968.6639 2Root[x3−3x2−4x+4,3]2+2Root[x3−4x2−2x+4,3]2+42 \text{Root}\left[x^3-3 x^2-4 x+4,3\right]^2+2 \text{Root}\left[x^3-4 x^2-2 x+4,3\right]^2+4

Characters

The symbolic character table is the following

123456871111Root[x3−3x2−4x+4,3]Root[x3−3x2−4x+4,3]Root[x3−4x2−2x+4,3]Root[x3−4x2−2x+4,3]1111Root[x3−3x2−4x+4,2]Root[x3−3x2−4x+4,2]Root[x3−4x2−2x+4,1]Root[x3−4x2−2x+4,1]1111Root[x3−3x2−4x+4,1]Root[x3−3x2−4x+4,1]Root[x3−4x2−2x+4,2]Root[x3−4x2−2x+4,2]11−1−100001−11−12−2001−11−1−11001−1−1100i2−i21−1−1100−i2i2\begin{array}{|cccccccc|} \hline \mathbf{1} & \mathbf{2} & \mathbf{3} & \mathbf{4} & \mathbf{5} & \mathbf{6} & \mathbf{8} & \mathbf{7} \\ \hline 1 & 1 & 1 & 1 & \text{Root}\left[x^3-3 x^2-4 x+4,3\right] & \text{Root}\left[x^3-3 x^2-4 x+4,3\right] & \text{Root}\left[x^3-4 x^2-2 x+4,3\right] & \text{Root}\left[x^3-4 x^2-2 x+4,3\right] \\ 1 & 1 & 1 & 1 & \text{Root}\left[x^3-3 x^2-4 x+4,2\right] & \text{Root}\left[x^3-3 x^2-4 x+4,2\right] & \text{Root}\left[x^3-4 x^2-2 x+4,1\right] & \text{Root}\left[x^3-4 x^2-2 x+4,1\right] \\ 1 & 1 & 1 & 1 & \text{Root}\left[x^3-3 x^2-4 x+4,1\right] & \text{Root}\left[x^3-3 x^2-4 x+4,1\right] & \text{Root}\left[x^3-4 x^2-2 x+4,2\right] & \text{Root}\left[x^3-4 x^2-2 x+4,2\right] \\ 1 & 1 & -1 & -1 & 0 & 0 & 0 & 0 \\ 1 & -1 & 1 & -1 & 2 & -2 & 0 & 0 \\ 1 & -1 & 1 & -1 & -1 & 1 & 0 & 0 \\ 1 & -1 & -1 & 1 & 0 & 0 & i \sqrt{2} & -i \sqrt{2} \\ 1 & -1 & -1 & 1 & 0 & 0 & -i \sqrt{2} & i \sqrt{2} \\ \hline \end{array}

The numeric character table is the following

123456871.0001.0001.0001.0003.7783.7784.2494.2491.0001.0001.0001.0000.71080.7108−1.103−1.1031.0001.0001.0001.000−1.489−1.4890.85360.85361.0001.000−1.000−1.00000001.000−1.0001.000−1.0002.000−2.000001.000−1.0001.000−1.000−1.0001.000001.000−1.000−1.0001.000001.414i−1.414i1.000−1.000−1.0001.00000−1.414i1.414i\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 & 1.000 & 1.000 & 3.778 & 3.778 & 4.249 & 4.249 \\ 1.000 & 1.000 & 1.000 & 1.000 & 0.7108 & 0.7108 & -1.103 & -1.103 \\ 1.000 & 1.000 & 1.000 & 1.000 & -1.489 & -1.489 & 0.8536 & 0.8536 \\ 1.000 & 1.000 & -1.000 & -1.000 & 0 & 0 & 0 & 0 \\ 1.000 & -1.000 & 1.000 & -1.000 & 2.000 & -2.000 & 0 & 0 \\ 1.000 & -1.000 & 1.000 & -1.000 & -1.000 & 1.000 & 0 & 0 \\ 1.000 & -1.000 & -1.000 & 1.000 & 0 & 0 & 1.414 i & -1.414 i \\ 1.000 & -1.000 & -1.000 & 1.000 & 0 & 0 & -1.414 i & 1.414 i \\ \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 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 extended cyclotomic criterion.

Data

Download links for numeric data: