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FR148,4\text{FR}^{8,4}_{14}

Fusion Rules

1234567821436587342178654312875656781+6+7+82+5+7+83+5+6+84+5+6+765872+5+7+81+6+7+84+5+6+73+5+6+878653+5+6+84+5+6+72+5+7+81+6+7+887564+5+6+73+5+6+81+6+7+82+5+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{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{7} & \mathbf{8} & \mathbf{1}+\mathbf{6}+\mathbf{7}+\mathbf{8} & \mathbf{2}+\mathbf{5}+\mathbf{7}+\mathbf{8} & \mathbf{3}+\mathbf{5}+\mathbf{6}+\mathbf{8} & \mathbf{4}+\mathbf{5}+\mathbf{6}+\mathbf{7} \\ \mathbf{6} & \mathbf{5} & \mathbf{8} & \mathbf{7} & \mathbf{2}+\mathbf{5}+\mathbf{7}+\mathbf{8} & \mathbf{1}+\mathbf{6}+\mathbf{7}+\mathbf{8} & \mathbf{4}+\mathbf{5}+\mathbf{6}+\mathbf{7} & \mathbf{3}+\mathbf{5}+\mathbf{6}+\mathbf{8} \\ \mathbf{7} & \mathbf{8} & \mathbf{6} & \mathbf{5} & \mathbf{3}+\mathbf{5}+\mathbf{6}+\mathbf{8} & \mathbf{4}+\mathbf{5}+\mathbf{6}+\mathbf{7} & \mathbf{2}+\mathbf{5}+\mathbf{7}+\mathbf{8} & \mathbf{1}+\mathbf{6}+\mathbf{7}+\mathbf{8} \\ \mathbf{8} & \mathbf{7} & \mathbf{5} & \mathbf{6} & \mathbf{4}+\mathbf{5}+\mathbf{6}+\mathbf{7} & \mathbf{3}+\mathbf{5}+\mathbf{6}+\mathbf{8} & \mathbf{1}+\mathbf{6}+\mathbf{7}+\mathbf{8} & \mathbf{2}+\mathbf{5}+\mathbf{7}+\mathbf{8} \\ \hline \end{array}

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

{(3 4)(7 8)}\{(\mathbf{3} \ \mathbf{4}) (\mathbf{7} \ \mathbf{8})\}

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}
{1,2,3,4}\{\mathbf{1},\mathbf{2},\mathbf{3},\mathbf{4}\} Z4: FR14,2\mathbb{Z}_4:\ \text{FR}^{4,2}_{1}

Quantum 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} 3.302783.30278 12(3+13)\frac{1}{2} \left(3+\sqrt{13}\right)
6\mathbf{6} 3.302783.30278 12(3+13)\frac{1}{2} \left(3+\sqrt{13}\right)
7\mathbf{7} 3.302783.30278 12(3+13)\frac{1}{2} \left(3+\sqrt{13}\right)
8\mathbf{8} 3.302783.30278 12(3+13)\frac{1}{2} \left(3+\sqrt{13}\right)
DFP2\mathcal{D}_{FP}^2 47.633347.6333 4+(3+13)24+\left(3+\sqrt{13}\right)^2

Characters

The symbolic character table is the following

12345876111112(3+13)12(3+13)12(3+13)12(3+13)111112(3−13)12(3−13)12(3−13)12(3−13)11−1−112(−1−5)12(1+5)12(1+5)12(−1−5)11−1−112(5−1)12(1−5)12(1−5)12(5−1)1−1−ii12(5−1)Root[x4+3x2+1,2]Root[x4+3x2+1,1]12(1−5)1−1i−i12(5−1)Root[x4+3x2+1,1]Root[x4+3x2+1,2]12(1−5)1−1i−i12(−1−5)Root[x4+3x2+1,4]Root[x4+3x2+1,3]12(1+5)1−1−ii12(−1−5)Root[x4+3x2+1,3]Root[x4+3x2+1,4]12(1+5)\begin{array}{|cccccccc|} \hline \mathbf{1} & \mathbf{2} & \mathbf{3} & \mathbf{4} & \mathbf{5} & \mathbf{8} & \mathbf{7} & \mathbf{6} \\ \hline 1 & 1 & 1 & 1 & \frac{1}{2} \left(3+\sqrt{13}\right) & \frac{1}{2} \left(3+\sqrt{13}\right) & \frac{1}{2} \left(3+\sqrt{13}\right) & \frac{1}{2} \left(3+\sqrt{13}\right) \\ 1 & 1 & 1 & 1 & \frac{1}{2} \left(3-\sqrt{13}\right) & \frac{1}{2} \left(3-\sqrt{13}\right) & \frac{1}{2} \left(3-\sqrt{13}\right) & \frac{1}{2} \left(3-\sqrt{13}\right) \\ 1 & 1 & -1 & -1 & \frac{1}{2} \left(-1-\sqrt{5}\right) & \frac{1}{2} \left(1+\sqrt{5}\right) & \frac{1}{2} \left(1+\sqrt{5}\right) & \frac{1}{2} \left(-1-\sqrt{5}\right) \\ 1 & 1 & -1 & -1 & \frac{1}{2} \left(\sqrt{5}-1\right) & \frac{1}{2} \left(1-\sqrt{5}\right) & \frac{1}{2} \left(1-\sqrt{5}\right) & \frac{1}{2} \left(\sqrt{5}-1\right) \\ 1 & -1 & -i & i & \frac{1}{2} \left(\sqrt{5}-1\right) & \text{Root}\left[x^4+3 x^2+1,2\right] & \text{Root}\left[x^4+3 x^2+1,1\right] & \frac{1}{2} \left(1-\sqrt{5}\right) \\ 1 & -1 & i & -i & \frac{1}{2} \left(\sqrt{5}-1\right) & \text{Root}\left[x^4+3 x^2+1,1\right] & \text{Root}\left[x^4+3 x^2+1,2\right] & \frac{1}{2} \left(1-\sqrt{5}\right) \\ 1 & -1 & i & -i & \frac{1}{2} \left(-1-\sqrt{5}\right) & \text{Root}\left[x^4+3 x^2+1,4\right] & \text{Root}\left[x^4+3 x^2+1,3\right] & \frac{1}{2} \left(1+\sqrt{5}\right) \\ 1 & -1 & -i & i & \frac{1}{2} \left(-1-\sqrt{5}\right) & \text{Root}\left[x^4+3 x^2+1,3\right] & \text{Root}\left[x^4+3 x^2+1,4\right] & \frac{1}{2} \left(1+\sqrt{5}\right) \\ \hline \end{array}

The numeric character table is the following

123458761.0001.0001.0001.0003.3033.3033.3033.3031.0001.0001.0001.000−0.3028−0.3028−0.3028−0.30281.0001.000−1.000−1.000−1.6181.6181.618−1.6181.0001.000−1.000−1.0000.6180−0.6180−0.61800.61801.000−1.000−1.000i1.000i0.61800ˋˋ4.3595026380819695+0.6180i0ˋˋ4.3595026380819695−0.6180i−0.61801.000−1.0001.000i−1.000i0.61800ˋˋ4.35950263808197−0.6180i0ˋˋ4.35950263808197+0.6180i−0.61801.000−1.0001.000i−1.000i−1.6180ˋˋ3.9415273575820122+1.618i0ˋˋ3.9415273575820122−1.618i1.6181.000−1.000−1.000i1.000i−1.6180ˋˋ3.9415273575820122−1.618i0ˋˋ3.9415273575820122+1.618i1.618\begin{array}{|rrrrrrrr|} \hline \mathbf{1} & \mathbf{2} & \mathbf{3} & \mathbf{4} & \mathbf{5} & \mathbf{8} & \mathbf{7} & \mathbf{6} \\ \hline 1.000 & 1.000 & 1.000 & 1.000 & 3.303 & 3.303 & 3.303 & 3.303 \\ 1.000 & 1.000 & 1.000 & 1.000 & -0.3028 & -0.3028 & -0.3028 & -0.3028 \\ 1.000 & 1.000 & -1.000 & -1.000 & -1.618 & 1.618 & 1.618 & -1.618 \\ 1.000 & 1.000 & -1.000 & -1.000 & 0.6180 & -0.6180 & -0.6180 & 0.6180 \\ 1.000 & -1.000 & -1.000 i & 1.000 i & 0.6180 & \text{0$\grave{ }\grave{ }$4.3595026380819695}+0.6180 i & \text{0$\grave{ }\grave{ }$4.3595026380819695}-0.6180 i & -0.6180 \\ 1.000 & -1.000 & 1.000 i & -1.000 i & 0.6180 & \text{0$\grave{ }\grave{ }$4.35950263808197}-0.6180 i & \text{0$\grave{ }\grave{ }$4.35950263808197}+0.6180 i & -0.6180 \\ 1.000 & -1.000 & 1.000 i & -1.000 i & -1.618 & \text{0$\grave{ }\grave{ }$3.9415273575820122}+1.618 i & \text{0$\grave{ }\grave{ }$3.9415273575820122}-1.618 i & 1.618 \\ 1.000 & -1.000 & -1.000 i & 1.000 i & -1.618 & \text{0$\grave{ }\grave{ }$3.9415273575820122}-1.618 i & \text{0$\grave{ }\grave{ }$3.9415273575820122}+1.618 i & 1.618 \\ \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 it is quadratic (i.e. its fusion categories are pivotal) and does not satisfy the pivotal version of the Drinfeld center criterion.

Data

Download links for numeric data: