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FR348,0\text{FR}^{8,0}_{34}

Fusion Rules

1234567821345687331+2+3+43+4+54+5+65+7+86+7+86+7+8443+4+51+2+3+5+63+4+7+84+6+7+85+6+7+85+6+7+8554+5+63+4+7+81+2+3+6+7+83+5+6+7+84+5+6+7+84+5+6+7+8665+7+84+6+7+83+5+6+7+81+2+4+5+6+7+83+4+5+6+7+83+4+5+6+7+8786+7+85+6+7+84+5+6+7+83+4+5+6+7+81+3+4+5+6+7+82+3+4+5+6+7+8876+7+85+6+7+84+5+6+7+83+4+5+6+7+82+3+4+5+6+7+81+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{4} & \mathbf{3}+\mathbf{4}+\mathbf{5} & \mathbf{4}+\mathbf{5}+\mathbf{6} & \mathbf{5}+\mathbf{7}+\mathbf{8} & \mathbf{6}+\mathbf{7}+\mathbf{8} & \mathbf{6}+\mathbf{7}+\mathbf{8} \\ \mathbf{4} & \mathbf{4} & \mathbf{3}+\mathbf{4}+\mathbf{5} & \mathbf{1}+\mathbf{2}+\mathbf{3}+\mathbf{5}+\mathbf{6} & \mathbf{3}+\mathbf{4}+\mathbf{7}+\mathbf{8} & \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{5} & \mathbf{5} & \mathbf{4}+\mathbf{5}+\mathbf{6} & \mathbf{3}+\mathbf{4}+\mathbf{7}+\mathbf{8} & \mathbf{1}+\mathbf{2}+\mathbf{3}+\mathbf{6}+\mathbf{7}+\mathbf{8} & \mathbf{3}+\mathbf{5}+\mathbf{6}+\mathbf{7}+\mathbf{8} & \mathbf{4}+\mathbf{5}+\mathbf{6}+\mathbf{7}+\mathbf{8} & \mathbf{4}+\mathbf{5}+\mathbf{6}+\mathbf{7}+\mathbf{8} \\ \mathbf{6} & \mathbf{6} & \mathbf{5}+\mathbf{7}+\mathbf{8} & \mathbf{4}+\mathbf{6}+\mathbf{7}+\mathbf{8} & \mathbf{3}+\mathbf{5}+\mathbf{6}+\mathbf{7}+\mathbf{8} & \mathbf{1}+\mathbf{2}+\mathbf{4}+\mathbf{5}+\mathbf{6}+\mathbf{7}+\mathbf{8} & \mathbf{3}+\mathbf{4}+\mathbf{5}+\mathbf{6}+\mathbf{7}+\mathbf{8} & \mathbf{3}+\mathbf{4}+\mathbf{5}+\mathbf{6}+\mathbf{7}+\mathbf{8} \\ \mathbf{7} & \mathbf{8} & \mathbf{6}+\mathbf{7}+\mathbf{8} & \mathbf{5}+\mathbf{6}+\mathbf{7}+\mathbf{8} & \mathbf{4}+\mathbf{5}+\mathbf{6}+\mathbf{7}+\mathbf{8} & \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{2}+\mathbf{3}+\mathbf{4}+\mathbf{5}+\mathbf{6}+\mathbf{7}+\mathbf{8} \\ \mathbf{8} & \mathbf{7} & \mathbf{6}+\mathbf{7}+\mathbf{8} & \mathbf{5}+\mathbf{6}+\mathbf{7}+\mathbf{8} & \mathbf{4}+\mathbf{5}+\mathbf{6}+\mathbf{7}+\mathbf{8} & \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} & \mathbf{1}+\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)}\{(\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}

Frobenius-Perron Dimensions

Particle Numeric Symbolic
1\mathbf{1} 1.1. 11
2\mathbf{2} 1.1. 11
3\mathbf{3} 2.964322.96432 Root[x6−5x5+2x4+17x3−10x2−16x+4,6]\text{Root}\left[x^6-5 x^5+2 x^4+17 x^3-10 x^2-16 x+4,6\right]
4\mathbf{4} 3.822863.82286 Root[x6−4x5−5x4+23x3−20x+4,6]\text{Root}\left[x^6-4 x^5-5 x^4+23 x^3-20 x+4,6\right]
5\mathbf{5} 4.544994.54499 Root[x6−4x5−7x4+21x3+2x2−20x+8,6]\text{Root}\left[x^6-4 x^5-7 x^4+21 x^3+2 x^2-20 x+8,6\right]
6\mathbf{6} 5.104955.10495 Root[x6−5x5−4x4+17x3+6x2−12x−4,6]\text{Root}\left[x^6-5 x^5-4 x^4+17 x^3+6 x^2-12 x-4,6\right]
7\mathbf{7} 5.293855.29385 Root[x6−4x5−8x4+5x3+6x2−x−1,6]\text{Root}\left[x^6-4 x^5-8 x^4+5 x^3+6 x^2-x-1,6\right]
8\mathbf{8} 5.293855.29385 Root[x6−4x5−8x4+5x3+6x2−x−1,6]\text{Root}\left[x^6-4 x^5-8 x^4+5 x^3+6 x^2-x-1,6\right]
DFP2\mathcal{D}_{FP}^2 128.169128.169 Root[x6−4x5−5x4+23x3−20x+4,6]2+Root[x6−5x5+2x4+17x3−10x2−16x+4,6]2+Root[x6−4x5−7x4+21x3+2x2−20x+8,6]2+Root[x6−5x5−4x4+17x3+6x2−12x−4,6]2+2Root[x6−4x5−8x4+5x3+6x2−x−1,6]2+2\text{Root}\left[x^6-4 x^5-5 x^4+23 x^3-20 x+4,6\right]^2+\text{Root}\left[x^6-5 x^5+2 x^4+17 x^3-10 x^2-16 x+4,6\right]^2+\text{Root}\left[x^6-4 x^5-7 x^4+21 x^3+2 x^2-20 x+8,6\right]^2+\text{Root}\left[x^6-5 x^5-4 x^4+17 x^3+6 x^2-12 x-4,6\right]^2+2 \text{Root}\left[x^6-4 x^5-8 x^4+5 x^3+6 x^2-x-1,6\right]^2+2

Characters

The symbolic character table is the following

1234568711Root[x6−5x5+2x4+17x3−10x2−16x+4,6]Root[x6−4x5−5x4+23x3−20x+4,6]Root[x6−4x5−7x4+21x3+2x2−20x+8,6]Root[x6−5x5−4x4+17x3+6x2−12x−4,6]Root[x6−4x5−8x4+5x3+6x2−x−1,6]Root[x6−4x5−8x4+5x3+6x2−x−1,6]11Root[x6−5x5+2x4+17x3−10x2−16x+4,5]Root[x6−4x5−5x4+23x3−20x+4,5]Root[x6−4x5−7x4+21x3+2x2−20x+8,4]Root[x6−5x5−4x4+17x3+6x2−12x−4,2]Root[x6−4x5−8x4+5x3+6x2−x−1,1]Root[x6−4x5−8x4+5x3+6x2−x−1,1]11Root[x6−5x5+2x4+17x3−10x2−16x+4,4]Root[x6−4x5−5x4+23x3−20x+4,2]Root[x6−4x5−7x4+21x3+2x2−20x+8,1]Root[x6−5x5−4x4+17x3+6x2−12x−4,3]Root[x6−4x5−8x4+5x3+6x2−x−1,5]Root[x6−4x5−8x4+5x3+6x2−x−1,5]11Root[x6−5x5+2x4+17x3−10x2−16x+4,3]Root[x6−4x5−5x4+23x3−20x+4,1]Root[x6−4x5−7x4+21x3+2x2−20x+8,5]Root[x6−5x5−4x4+17x3+6x2−12x−4,4]Root[x6−4x5−8x4+5x3+6x2−x−1,2]Root[x6−4x5−8x4+5x3+6x2−x−1,2]11Root[x6−5x5+2x4+17x3−10x2−16x+4,2]Root[x6−4x5−5x4+23x3−20x+4,3]Root[x6−4x5−7x4+21x3+2x2−20x+8,3]Root[x6−5x5−4x4+17x3+6x2−12x−4,1]Root[x6−4x5−8x4+5x3+6x2−x−1,4]Root[x6−4x5−8x4+5x3+6x2−x−1,4]11Root[x6−5x5+2x4+17x3−10x2−16x+4,1]Root[x6−4x5−5x4+23x3−20x+4,4]Root[x6−4x5−7x4+21x3+2x2−20x+8,2]Root[x6−5x5−4x4+17x3+6x2−12x−4,5]Root[x6−4x5−8x4+5x3+6x2−x−1,3]Root[x6−4x5−8x4+5x3+6x2−x−1,3]1−100001−11−10000−11\begin{array}{|cccccccc|} \hline \mathbf{1} & \mathbf{2} & \mathbf{3} & \mathbf{4} & \mathbf{5} & \mathbf{6} & \mathbf{8} & \mathbf{7} \\ \hline 1 & 1 & \text{Root}\left[x^6-5 x^5+2 x^4+17 x^3-10 x^2-16 x+4,6\right] & \text{Root}\left[x^6-4 x^5-5 x^4+23 x^3-20 x+4,6\right] & \text{Root}\left[x^6-4 x^5-7 x^4+21 x^3+2 x^2-20 x+8,6\right] & \text{Root}\left[x^6-5 x^5-4 x^4+17 x^3+6 x^2-12 x-4,6\right] & \text{Root}\left[x^6-4 x^5-8 x^4+5 x^3+6 x^2-x-1,6\right] & \text{Root}\left[x^6-4 x^5-8 x^4+5 x^3+6 x^2-x-1,6\right] \\ 1 & 1 & \text{Root}\left[x^6-5 x^5+2 x^4+17 x^3-10 x^2-16 x+4,5\right] & \text{Root}\left[x^6-4 x^5-5 x^4+23 x^3-20 x+4,5\right] & \text{Root}\left[x^6-4 x^5-7 x^4+21 x^3+2 x^2-20 x+8,4\right] & \text{Root}\left[x^6-5 x^5-4 x^4+17 x^3+6 x^2-12 x-4,2\right] & \text{Root}\left[x^6-4 x^5-8 x^4+5 x^3+6 x^2-x-1,1\right] & \text{Root}\left[x^6-4 x^5-8 x^4+5 x^3+6 x^2-x-1,1\right] \\ 1 & 1 & \text{Root}\left[x^6-5 x^5+2 x^4+17 x^3-10 x^2-16 x+4,4\right] & \text{Root}\left[x^6-4 x^5-5 x^4+23 x^3-20 x+4,2\right] & \text{Root}\left[x^6-4 x^5-7 x^4+21 x^3+2 x^2-20 x+8,1\right] & \text{Root}\left[x^6-5 x^5-4 x^4+17 x^3+6 x^2-12 x-4,3\right] & \text{Root}\left[x^6-4 x^5-8 x^4+5 x^3+6 x^2-x-1,5\right] & \text{Root}\left[x^6-4 x^5-8 x^4+5 x^3+6 x^2-x-1,5\right] \\ 1 & 1 & \text{Root}\left[x^6-5 x^5+2 x^4+17 x^3-10 x^2-16 x+4,3\right] & \text{Root}\left[x^6-4 x^5-5 x^4+23 x^3-20 x+4,1\right] & \text{Root}\left[x^6-4 x^5-7 x^4+21 x^3+2 x^2-20 x+8,5\right] & \text{Root}\left[x^6-5 x^5-4 x^4+17 x^3+6 x^2-12 x-4,4\right] & \text{Root}\left[x^6-4 x^5-8 x^4+5 x^3+6 x^2-x-1,2\right] & \text{Root}\left[x^6-4 x^5-8 x^4+5 x^3+6 x^2-x-1,2\right] \\ 1 & 1 & \text{Root}\left[x^6-5 x^5+2 x^4+17 x^3-10 x^2-16 x+4,2\right] & \text{Root}\left[x^6-4 x^5-5 x^4+23 x^3-20 x+4,3\right] & \text{Root}\left[x^6-4 x^5-7 x^4+21 x^3+2 x^2-20 x+8,3\right] & \text{Root}\left[x^6-5 x^5-4 x^4+17 x^3+6 x^2-12 x-4,1\right] & \text{Root}\left[x^6-4 x^5-8 x^4+5 x^3+6 x^2-x-1,4\right] & \text{Root}\left[x^6-4 x^5-8 x^4+5 x^3+6 x^2-x-1,4\right] \\ 1 & 1 & \text{Root}\left[x^6-5 x^5+2 x^4+17 x^3-10 x^2-16 x+4,1\right] & \text{Root}\left[x^6-4 x^5-5 x^4+23 x^3-20 x+4,4\right] & \text{Root}\left[x^6-4 x^5-7 x^4+21 x^3+2 x^2-20 x+8,2\right] & \text{Root}\left[x^6-5 x^5-4 x^4+17 x^3+6 x^2-12 x-4,5\right] & \text{Root}\left[x^6-4 x^5-8 x^4+5 x^3+6 x^2-x-1,3\right] & \text{Root}\left[x^6-4 x^5-8 x^4+5 x^3+6 x^2-x-1,3\right] \\ 1 & -1 & 0 & 0 & 0 & 0 & 1 & -1 \\ 1 & -1 & 0 & 0 & 0 & 0 & -1 & 1 \\ \hline \end{array}

The numeric character table is the following

123456871.0001.0002.9643.8234.5455.1055.2945.2941.0001.0002.5872.1040.7510−0.9122−1.555−1.5551.0001.0001.608−1.023−2.229−0.33240.84750.84751.0001.0000.2300−2.1771.4461.063−0.6008−0.60081.0001.000−1.0680.21010.6339−1.5210.49580.49581.0001.000−1.3201.063−1.1471.597−0.4811−0.48111.000−1.00000001.000−1.0001.000−1.0000000−1.0001.000\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 & 2.964 & 3.823 & 4.545 & 5.105 & 5.294 & 5.294 \\ 1.000 & 1.000 & 2.587 & 2.104 & 0.7510 & -0.9122 & -1.555 & -1.555 \\ 1.000 & 1.000 & 1.608 & -1.023 & -2.229 & -0.3324 & 0.8475 & 0.8475 \\ 1.000 & 1.000 & 0.2300 & -2.177 & 1.446 & 1.063 & -0.6008 & -0.6008 \\ 1.000 & 1.000 & -1.068 & 0.2101 & 0.6339 & -1.521 & 0.4958 & 0.4958 \\ 1.000 & 1.000 & -1.320 & 1.063 & -1.147 & 1.597 & -0.4811 & -0.4811 \\ 1.000 & -1.000 & 0 & 0 & 0 & 0 & 1.000 & -1.000 \\ 1.000 & -1.000 & 0 & 0 & 0 & 0 & -1.000 & 1.000 \\ \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 zero spectrum criterion.

Data

Download links for numeric data: