AnyonWiki

FR149,6\text{FR}^{9,6}_{14}

Fusion Rules

12345678921435987634215796843125869755551+2+3+4+6+7+8+95+6+7+8+95+6+7+8+95+6+7+8+95+6+7+8+969785+6+7+8+94+5+6+7+8+91+5+6+7+8+92+5+6+7+8+93+5+6+7+8+978965+6+7+8+91+5+6+7+8+93+5+6+7+8+94+5+6+7+8+92+5+6+7+8+987695+6+7+8+92+5+6+7+8+94+5+6+7+8+93+5+6+7+8+91+5+6+7+8+996875+6+7+8+93+5+6+7+8+92+5+6+7+8+91+5+6+7+8+94+5+6+7+8+9\begin{array}{|lllllllll|} \hline \mathbf{1} & \mathbf{2} & \mathbf{3} & \mathbf{4} & \mathbf{5} & \mathbf{6} & \mathbf{7} & \mathbf{8} & \mathbf{9} \\ \mathbf{2} & \mathbf{1} & \mathbf{4} & \mathbf{3} & \mathbf{5} & \mathbf{9} & \mathbf{8} & \mathbf{7} & \mathbf{6} \\ \mathbf{3} & \mathbf{4} & \mathbf{2} & \mathbf{1} & \mathbf{5} & \mathbf{7} & \mathbf{9} & \mathbf{6} & \mathbf{8} \\ \mathbf{4} & \mathbf{3} & \mathbf{1} & \mathbf{2} & \mathbf{5} & \mathbf{8} & \mathbf{6} & \mathbf{9} & \mathbf{7} \\ \mathbf{5} & \mathbf{5} & \mathbf{5} & \mathbf{5} & \mathbf{1}+\mathbf{2}+\mathbf{3}+\mathbf{4}+\mathbf{6}+\mathbf{7}+\mathbf{8}+\mathbf{9} & \mathbf{5}+\mathbf{6}+\mathbf{7}+\mathbf{8}+\mathbf{9} & \mathbf{5}+\mathbf{6}+\mathbf{7}+\mathbf{8}+\mathbf{9} & \mathbf{5}+\mathbf{6}+\mathbf{7}+\mathbf{8}+\mathbf{9} & \mathbf{5}+\mathbf{6}+\mathbf{7}+\mathbf{8}+\mathbf{9} \\ \mathbf{6} & \mathbf{9} & \mathbf{7} & \mathbf{8} & \mathbf{5}+\mathbf{6}+\mathbf{7}+\mathbf{8}+\mathbf{9} & \mathbf{4}+\mathbf{5}+\mathbf{6}+\mathbf{7}+\mathbf{8}+\mathbf{9} & \mathbf{1}+\mathbf{5}+\mathbf{6}+\mathbf{7}+\mathbf{8}+\mathbf{9} & \mathbf{2}+\mathbf{5}+\mathbf{6}+\mathbf{7}+\mathbf{8}+\mathbf{9} & \mathbf{3}+\mathbf{5}+\mathbf{6}+\mathbf{7}+\mathbf{8}+\mathbf{9} \\ \mathbf{7} & \mathbf{8} & \mathbf{9} & \mathbf{6} & \mathbf{5}+\mathbf{6}+\mathbf{7}+\mathbf{8}+\mathbf{9} & \mathbf{1}+\mathbf{5}+\mathbf{6}+\mathbf{7}+\mathbf{8}+\mathbf{9} & \mathbf{3}+\mathbf{5}+\mathbf{6}+\mathbf{7}+\mathbf{8}+\mathbf{9} & \mathbf{4}+\mathbf{5}+\mathbf{6}+\mathbf{7}+\mathbf{8}+\mathbf{9} & \mathbf{2}+\mathbf{5}+\mathbf{6}+\mathbf{7}+\mathbf{8}+\mathbf{9} \\ \mathbf{8} & \mathbf{7} & \mathbf{6} & \mathbf{9} & \mathbf{5}+\mathbf{6}+\mathbf{7}+\mathbf{8}+\mathbf{9} & \mathbf{2}+\mathbf{5}+\mathbf{6}+\mathbf{7}+\mathbf{8}+\mathbf{9} & \mathbf{4}+\mathbf{5}+\mathbf{6}+\mathbf{7}+\mathbf{8}+\mathbf{9} & \mathbf{3}+\mathbf{5}+\mathbf{6}+\mathbf{7}+\mathbf{8}+\mathbf{9} & \mathbf{1}+\mathbf{5}+\mathbf{6}+\mathbf{7}+\mathbf{8}+\mathbf{9} \\ \mathbf{9} & \mathbf{6} & \mathbf{8} & \mathbf{7} & \mathbf{5}+\mathbf{6}+\mathbf{7}+\mathbf{8}+\mathbf{9} & \mathbf{3}+\mathbf{5}+\mathbf{6}+\mathbf{7}+\mathbf{8}+\mathbf{9} & \mathbf{2}+\mathbf{5}+\mathbf{6}+\mathbf{7}+\mathbf{8}+\mathbf{9} & \mathbf{1}+\mathbf{5}+\mathbf{6}+\mathbf{7}+\mathbf{8}+\mathbf{9} & \mathbf{4}+\mathbf{5}+\mathbf{6}+\mathbf{7}+\mathbf{8}+\mathbf{9} \\ \hline \end{array}

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

{(6 9)(7 8),(3 4)(6 7)(8 9)}\{(\mathbf{6} \ \mathbf{9}) (\mathbf{7} \ \mathbf{8}), (\mathbf{3} \ \mathbf{4}) (\mathbf{6} \ \mathbf{7}) (\mathbf{8} \ \mathbf{9})\}

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} 4.962394.96239 Root[x3−4x2−8x+16,3]\text{Root}\left[x^3-4 x^2-8 x+16,3\right]
6\mathbf{6} 5.156335.15633 Root[x3−5x2−x+1,3]\text{Root}\left[x^3-5 x^2-x+1,3\right]
7\mathbf{7} 5.156335.15633 Root[x3−5x2−x+1,3]\text{Root}\left[x^3-5 x^2-x+1,3\right]
8\mathbf{8} 5.156335.15633 Root[x3−5x2−x+1,3]\text{Root}\left[x^3-5 x^2-x+1,3\right]
9\mathbf{9} 5.156335.15633 Root[x3−5x2−x+1,3]\text{Root}\left[x^3-5 x^2-x+1,3\right]
DFP2\mathcal{D}_{FP}^2 134.976134.976 Root[x3−4x2−8x+16,3]2+4Root[x3−5x2−x+1,3]2+4\text{Root}\left[x^3-4 x^2-8 x+16,3\right]^2+4 \text{Root}\left[x^3-5 x^2-x+1,3\right]^2+4

Characters

The symbolic character table is the following

1243576981111Root[x3−4x2−8x+16,3]Root[x3−5x2−x+1,3]Root[x3−5x2−x+1,3]Root[x3−5x2−x+1,3]Root[x3−5x2−x+1,3]1111Root[x3−4x2−8x+16,2]Root[x3−5x2−x+1,1]Root[x3−5x2−x+1,1]Root[x3−5x2−x+1,1]Root[x3−5x2−x+1,1]1111Root[x3−4x2−8x+16,1]Root[x3−5x2−x+1,2]Root[x3−5x2−x+1,2]Root[x3−5x2−x+1,2]Root[x3−5x2−x+1,2]11−1−10−iii−i11−1−10i−i−ii1−1−ii0Root[x4+1,4]Root[x4+1,3]Root[x4+1,2]Root[x4+1,1]1−1i−i0Root[x4+1,3]Root[x4+1,4]Root[x4+1,1]Root[x4+1,2]1−1i−i0Root[x4+1,2]Root[x4+1,1]Root[x4+1,4]Root[x4+1,3]1−1−ii0Root[x4+1,1]Root[x4+1,2]Root[x4+1,3]Root[x4+1,4]\begin{array}{|ccccccccc|} \hline \mathbf{1} & \mathbf{2} & \mathbf{4} & \mathbf{3} & \mathbf{5} & \mathbf{7} & \mathbf{6} & \mathbf{9} & \mathbf{8} \\ \hline 1 & 1 & 1 & 1 & \text{Root}\left[x^3-4 x^2-8 x+16,3\right] & \text{Root}\left[x^3-5 x^2-x+1,3\right] & \text{Root}\left[x^3-5 x^2-x+1,3\right] & \text{Root}\left[x^3-5 x^2-x+1,3\right] & \text{Root}\left[x^3-5 x^2-x+1,3\right] \\ 1 & 1 & 1 & 1 & \text{Root}\left[x^3-4 x^2-8 x+16,2\right] & \text{Root}\left[x^3-5 x^2-x+1,1\right] & \text{Root}\left[x^3-5 x^2-x+1,1\right] & \text{Root}\left[x^3-5 x^2-x+1,1\right] & \text{Root}\left[x^3-5 x^2-x+1,1\right] \\ 1 & 1 & 1 & 1 & \text{Root}\left[x^3-4 x^2-8 x+16,1\right] & \text{Root}\left[x^3-5 x^2-x+1,2\right] & \text{Root}\left[x^3-5 x^2-x+1,2\right] & \text{Root}\left[x^3-5 x^2-x+1,2\right] & \text{Root}\left[x^3-5 x^2-x+1,2\right] \\ 1 & 1 & -1 & -1 & 0 & -i & i & i & -i \\ 1 & 1 & -1 & -1 & 0 & i & -i & -i & i \\ 1 & -1 & -i & i & 0 & \text{Root}\left[x^4+1,4\right] & \text{Root}\left[x^4+1,3\right] & \text{Root}\left[x^4+1,2\right] & \text{Root}\left[x^4+1,1\right] \\ 1 & -1 & i & -i & 0 & \text{Root}\left[x^4+1,3\right] & \text{Root}\left[x^4+1,4\right] & \text{Root}\left[x^4+1,1\right] & \text{Root}\left[x^4+1,2\right] \\ 1 & -1 & i & -i & 0 & \text{Root}\left[x^4+1,2\right] & \text{Root}\left[x^4+1,1\right] & \text{Root}\left[x^4+1,4\right] & \text{Root}\left[x^4+1,3\right] \\ 1 & -1 & -i & i & 0 & \text{Root}\left[x^4+1,1\right] & \text{Root}\left[x^4+1,2\right] & \text{Root}\left[x^4+1,3\right] & \text{Root}\left[x^4+1,4\right] \\ \hline \end{array}

The numeric character table is the following

1243576981.0001.0001.0001.0004.9625.1565.1565.1565.1561.0001.0001.0001.0001.378−0.5254−0.5254−0.5254−0.52541.0001.0001.0001.000−2.3400.36910.36910.36910.36911.0001.000−1.000−1.0000−1.000i1.000i1.000i−1.000i1.0001.000−1.000−1.00001.000i−1.000i−1.000i1.000i1.000−1.000−1.000i1.000i00.7071+0.7071i0.7071−0.7071i−0.7071+0.7071i−0.7071−0.7071i1.000−1.0001.000i−1.000i00.7071−0.7071i0.7071+0.7071i−0.7071−0.7071i−0.7071+0.7071i1.000−1.0001.000i−1.000i0−0.7071+0.7071i−0.7071−0.7071i0.7071+0.7071i0.7071−0.7071i1.000−1.000−1.000i1.000i0−0.7071−0.7071i−0.7071+0.7071i0.7071−0.7071i0.7071+0.7071i\begin{array}{|rrrrrrrrr|} \hline \mathbf{1} & \mathbf{2} & \mathbf{4} & \mathbf{3} & \mathbf{5} & \mathbf{7} & \mathbf{6} & \mathbf{9} & \mathbf{8} \\ \hline 1.000 & 1.000 & 1.000 & 1.000 & 4.962 & 5.156 & 5.156 & 5.156 & 5.156 \\ 1.000 & 1.000 & 1.000 & 1.000 & 1.378 & -0.5254 & -0.5254 & -0.5254 & -0.5254 \\ 1.000 & 1.000 & 1.000 & 1.000 & -2.340 & 0.3691 & 0.3691 & 0.3691 & 0.3691 \\ 1.000 & 1.000 & -1.000 & -1.000 & 0 & -1.000 i & 1.000 i & 1.000 i & -1.000 i \\ 1.000 & 1.000 & -1.000 & -1.000 & 0 & 1.000 i & -1.000 i & -1.000 i & 1.000 i \\ 1.000 & -1.000 & -1.000 i & 1.000 i & 0 & 0.7071+0.7071 i & 0.7071-0.7071 i & -0.7071+0.7071 i & -0.7071-0.7071 i \\ 1.000 & -1.000 & 1.000 i & -1.000 i & 0 & 0.7071-0.7071 i & 0.7071+0.7071 i & -0.7071-0.7071 i & -0.7071+0.7071 i \\ 1.000 & -1.000 & 1.000 i & -1.000 i & 0 & -0.7071+0.7071 i & -0.7071-0.7071 i & 0.7071+0.7071 i & 0.7071-0.7071 i \\ 1.000 & -1.000 & -1.000 i & 1.000 i & 0 & -0.7071-0.7071 i & -0.7071+0.7071 i & 0.7071-0.7071 i & 0.7071+0.7071 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: