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FR77,4\text{FR}^{7,4}_{7}

Fusion Rules

1234567231467531247564441+2+3+5+6+74+5+6+74+5+6+74+5+6+75674+5+6+71+4+5+6+72+4+5+6+73+4+5+6+76754+5+6+72+4+5+6+73+4+5+6+71+4+5+6+77564+5+6+73+4+5+6+71+4+5+6+72+4+5+6+7\begin{array}{|lllllll|} \hline \mathbf{1} & \mathbf{2} & \mathbf{3} & \mathbf{4} & \mathbf{5} & \mathbf{6} & \mathbf{7} \\ \mathbf{2} & \mathbf{3} & \mathbf{1} & \mathbf{4} & \mathbf{6} & \mathbf{7} & \mathbf{5} \\ \mathbf{3} & \mathbf{1} & \mathbf{2} & \mathbf{4} & \mathbf{7} & \mathbf{5} & \mathbf{6} \\ \mathbf{4} & \mathbf{4} & \mathbf{4} & \mathbf{1}+\mathbf{2}+\mathbf{3}+\mathbf{5}+\mathbf{6}+\mathbf{7} & \mathbf{4}+\mathbf{5}+\mathbf{6}+\mathbf{7} & \mathbf{4}+\mathbf{5}+\mathbf{6}+\mathbf{7} & \mathbf{4}+\mathbf{5}+\mathbf{6}+\mathbf{7} \\ \mathbf{5} & \mathbf{6} & \mathbf{7} & \mathbf{4}+\mathbf{5}+\mathbf{6}+\mathbf{7} & \mathbf{1}+\mathbf{4}+\mathbf{5}+\mathbf{6}+\mathbf{7} & \mathbf{2}+\mathbf{4}+\mathbf{5}+\mathbf{6}+\mathbf{7} & \mathbf{3}+\mathbf{4}+\mathbf{5}+\mathbf{6}+\mathbf{7} \\ \mathbf{6} & \mathbf{7} & \mathbf{5} & \mathbf{4}+\mathbf{5}+\mathbf{6}+\mathbf{7} & \mathbf{2}+\mathbf{4}+\mathbf{5}+\mathbf{6}+\mathbf{7} & \mathbf{3}+\mathbf{4}+\mathbf{5}+\mathbf{6}+\mathbf{7} & \mathbf{1}+\mathbf{4}+\mathbf{5}+\mathbf{6}+\mathbf{7} \\ \mathbf{7} & \mathbf{5} & \mathbf{6} & \mathbf{4}+\mathbf{5}+\mathbf{6}+\mathbf{7} & \mathbf{3}+\mathbf{4}+\mathbf{5}+\mathbf{6}+\mathbf{7} & \mathbf{1}+\mathbf{4}+\mathbf{5}+\mathbf{6}+\mathbf{7} & \mathbf{2}+\mathbf{4}+\mathbf{5}+\mathbf{6}+\mathbf{7} \\ \hline \end{array}

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

{(2 3)(6 7)}\{(\mathbf{2} \ \mathbf{3}) (\mathbf{6} \ \mathbf{7})\}

The following particles form non-trivial sub fusion rings

Particles SubRing
{1,2,3}\{\mathbf{1},\mathbf{2},\mathbf{3}\} Z3: FR13,2\mathbb{Z}_3:\ \text{FR}^{3,2}_{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} 3.942823.94282 Root[x3−3x2−6x+9,3]\text{Root}\left[x^3-3 x^2-6 x+9,3\right]
5\mathbf{5} 4.181944.18194 Root[x3−4x2−x+1,3]\text{Root}\left[x^3-4 x^2-x+1,3\right]
6\mathbf{6} 4.181944.18194 Root[x3−4x2−x+1,3]\text{Root}\left[x^3-4 x^2-x+1,3\right]
7\mathbf{7} 4.181944.18194 Root[x3−4x2−x+1,3]\text{Root}\left[x^3-4 x^2-x+1,3\right]
DFP2\mathcal{D}_{FP}^2 71.011871.0118 Root[x3−3x2−6x+9,3]2+3Root[x3−4x2−x+1,3]2+3\text{Root}\left[x^3-3 x^2-6 x+9,3\right]^2+3 \text{Root}\left[x^3-4 x^2-x+1,3\right]^2+3

Characters

The symbolic character table is the following

1234576111Root[x3−3x2−6x+9,3]Root[x3−4x2−x+1,3]Root[x3−4x2−x+1,3]Root[x3−4x2−x+1,3]111Root[x3−3x2−6x+9,2]Root[x3−4x2−x+1,1]Root[x3−4x2−x+1,1]Root[x3−4x2−x+1,1]111Root[x3−3x2−6x+9,1]Root[x3−4x2−x+1,2]Root[x3−4x2−x+1,2]Root[x3−4x2−x+1,2]10−101−101−10010−1112(−1+i3)12(−1−i3)0−112(1+i3)12(1−i3)112(−1−i3)12(−1+i3)0−112(1−i3)12(1+i3)\begin{array}{|ccccccc|} \hline \mathbf{1} & \mathbf{2} & \mathbf{3} & \mathbf{4} & \mathbf{5} & \mathbf{7} & \mathbf{6} \\ \hline 1 & 1 & 1 & \text{Root}\left[x^3-3 x^2-6 x+9,3\right] & \text{Root}\left[x^3-4 x^2-x+1,3\right] & \text{Root}\left[x^3-4 x^2-x+1,3\right] & \text{Root}\left[x^3-4 x^2-x+1,3\right] \\ 1 & 1 & 1 & \text{Root}\left[x^3-3 x^2-6 x+9,2\right] & \text{Root}\left[x^3-4 x^2-x+1,1\right] & \text{Root}\left[x^3-4 x^2-x+1,1\right] & \text{Root}\left[x^3-4 x^2-x+1,1\right] \\ 1 & 1 & 1 & \text{Root}\left[x^3-3 x^2-6 x+9,1\right] & \text{Root}\left[x^3-4 x^2-x+1,2\right] & \text{Root}\left[x^3-4 x^2-x+1,2\right] & \text{Root}\left[x^3-4 x^2-x+1,2\right] \\ 1 & 0 & -1 & 0 & 1 & -1 & 0 \\ 1 & -1 & 0 & 0 & 1 & 0 & -1 \\ 1 & \frac{1}{2} \left(-1+i \sqrt{3}\right) & \frac{1}{2} \left(-1-i \sqrt{3}\right) & 0 & -1 & \frac{1}{2} \left(1+i \sqrt{3}\right) & \frac{1}{2} \left(1-i \sqrt{3}\right) \\ 1 & \frac{1}{2} \left(-1-i \sqrt{3}\right) & \frac{1}{2} \left(-1+i \sqrt{3}\right) & 0 & -1 & \frac{1}{2} \left(1-i \sqrt{3}\right) & \frac{1}{2} \left(1+i \sqrt{3}\right) \\ \hline \end{array}

The numeric character table is the following

12345761.0001.0001.0003.9434.1824.1824.1821.0001.0001.0001.111−0.5884−0.5884−0.58841.0001.0001.000−2.0540.40640.40640.40641.0000−1.00001.000−1.00001.000−1.000001.0000−1.0001.000−0.5000+0.8660i−0.5000−0.8660i0−1.0000.5000+0.8660i0.5000−0.8660i1.000−0.5000−0.8660i−0.5000+0.8660i0−1.0000.5000−0.8660i0.5000+0.8660i\begin{array}{|rrrrrrr|} \hline \mathbf{1} & \mathbf{2} & \mathbf{3} & \mathbf{4} & \mathbf{5} & \mathbf{7} & \mathbf{6} \\ \hline 1.000 & 1.000 & 1.000 & 3.943 & 4.182 & 4.182 & 4.182 \\ 1.000 & 1.000 & 1.000 & 1.111 & -0.5884 & -0.5884 & -0.5884 \\ 1.000 & 1.000 & 1.000 & -2.054 & 0.4064 & 0.4064 & 0.4064 \\ 1.000 & 0 & -1.000 & 0 & 1.000 & -1.000 & 0 \\ 1.000 & -1.000 & 0 & 0 & 1.000 & 0 & -1.000 \\ 1.000 & -0.5000+0.8660 i & -0.5000-0.8660 i & 0 & -1.000 & 0.5000+0.8660 i & 0.5000-0.8660 i \\ 1.000 & -0.5000-0.8660 i & -0.5000+0.8660 i & 0 & -1.000 & 0.5000-0.8660 i & 0.5000+0.8660 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 extended cyclotomic criterion.

Data

Download links for numeric data: