AnyonWiki

FR79,4\text{FR}^{9,4}_{7}

Fusion Rules

1234567892165438793641257894513627895426318796352148797877881+3+4+92+5+6+97+8+98788772+5+6+91+3+4+97+8+99999997+8+97+8+91+2+3+4+5+6+7+8\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{6} & \mathbf{5} & \mathbf{4} & \mathbf{3} & \mathbf{8} & \mathbf{7} & \mathbf{9} \\ \mathbf{3} & \mathbf{6} & \mathbf{4} & \mathbf{1} & \mathbf{2} & \mathbf{5} & \mathbf{7} & \mathbf{8} & \mathbf{9} \\ \mathbf{4} & \mathbf{5} & \mathbf{1} & \mathbf{3} & \mathbf{6} & \mathbf{2} & \mathbf{7} & \mathbf{8} & \mathbf{9} \\ \mathbf{5} & \mathbf{4} & \mathbf{2} & \mathbf{6} & \mathbf{3} & \mathbf{1} & \mathbf{8} & \mathbf{7} & \mathbf{9} \\ \mathbf{6} & \mathbf{3} & \mathbf{5} & \mathbf{2} & \mathbf{1} & \mathbf{4} & \mathbf{8} & \mathbf{7} & \mathbf{9} \\ \mathbf{7} & \mathbf{8} & \mathbf{7} & \mathbf{7} & \mathbf{8} & \mathbf{8} & \mathbf{1}+\mathbf{3}+\mathbf{4}+\mathbf{9} & \mathbf{2}+\mathbf{5}+\mathbf{6}+\mathbf{9} & \mathbf{7}+\mathbf{8}+\mathbf{9} \\ \mathbf{8} & \mathbf{7} & \mathbf{8} & \mathbf{8} & \mathbf{7} & \mathbf{7} & \mathbf{2}+\mathbf{5}+\mathbf{6}+\mathbf{9} & \mathbf{1}+\mathbf{3}+\mathbf{4}+\mathbf{9} & \mathbf{7}+\mathbf{8}+\mathbf{9} \\ \mathbf{9} & \mathbf{9} & \mathbf{9} & \mathbf{9} & \mathbf{9} & \mathbf{9} & \mathbf{7}+\mathbf{8}+\mathbf{9} & \mathbf{7}+\mathbf{8}+\mathbf{9} & \mathbf{1}+\mathbf{2}+\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),(3 4)(5 6)}\{(\mathbf{7} \ \mathbf{8}), (\mathbf{3} \ \mathbf{4}) (\mathbf{5} \ \mathbf{6})\}

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}
{1,3,4}\{\mathbf{1},\mathbf{3},\mathbf{4}\} Z3: FR13,2\mathbb{Z}_3:\ \text{FR}^{3,2}_{1}
{1,2,3,4,5,6}\{\mathbf{1},\mathbf{2},\mathbf{3},\mathbf{4},\mathbf{5},\mathbf{6}\} Z6: FR16,4\mathbb{Z}_6:\ \text{FR}^{6,4}_{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} 1.1. 11
6\mathbf{6} 1.1. 11
7\mathbf{7} 2.514142.51414 Root[x3−x2−5x+3,3]\text{Root}\left[x^3-x^2-5 x+3,3\right]
8\mathbf{8} 2.514142.51414 Root[x3−x2−5x+3,3]\text{Root}\left[x^3-x^2-5 x+3,3\right]
9\mathbf{9} 3.320883.32088 Root[x3−2x2−8x+12,3]\text{Root}\left[x^3-2 x^2-8 x+12,3\right]
DFP2\mathcal{D}_{FP}^2 29.6729.67 2Root[x3−x2−5x+3,3]2+Root[x3−2x2−8x+12,3]2+62 \text{Root}\left[x^3-x^2-5 x+3,3\right]^2+\text{Root}\left[x^3-2 x^2-8 x+12,3\right]^2+6

Characters

The symbolic character table is the following

1243657891.1.1.1.1.1.2.514142.514143.320881.1.1.1.1.1.0.5719930.571993−2.672821.1.1.1.1.1.−2.08613−2.086131.351941.−1.1.1.−1.−1.−1.732051.732050.1.−1.1.1.−1.−1.1.73205−1.732050.1.1. +3.108331249142488ˋ*∧-115i−0.5+0.866025i−0.5−0.866025i−0.5−0.866025i−0.5+0.866025i0.0.0.1.1. −3.108331249142488ˋ*∧-115i−0.5−0.866025i−0.5+0.866025i−0.5+0.866025i−0.5−0.866025i0.0.0.1.−1.−6.838328748113474ˋ*∧-115i−0.5+0.866025i−0.5−0.866025i0.5 +0.866025i0.5 −0.866025i0.0.0.1.−1.+6.838328748113474ˋ*∧-115i−0.5−0.866025i−0.5+0.866025i0.5 −0.866025i0.5 +0.866025i0.0.0.\begin{array}{|ccccccccc|} \hline \mathbf{1} & \mathbf{2} & \mathbf{4} & \mathbf{3} & \mathbf{6} & \mathbf{5} & \mathbf{7} & \mathbf{8} & \mathbf{9} \\ \hline 1. & 1. & 1. & 1. & 1. & 1. & 2.51414 & 2.51414 & 3.32088 \\ 1. & 1. & 1. & 1. & 1. & 1. & 0.571993 & 0.571993 & -2.67282 \\ 1. & 1. & 1. & 1. & 1. & 1. & -2.08613 & -2.08613 & 1.35194 \\ 1. & -1. & 1. & 1. & -1. & -1. & -1.73205 & 1.73205 & 0. \\ 1. & -1. & 1. & 1. & -1. & -1. & 1.73205 & -1.73205 & 0. \\ 1. & 1.\, +\text{3.108331249142488$\grave{ }$*${}^{\wedge}$-115} i & -0.5+0.866025 i & -0.5-0.866025 i & -0.5-0.866025 i & -0.5+0.866025 i & 0. & 0. & 0. \\ 1. & 1.\, -\text{3.108331249142488$\grave{ }$*${}^{\wedge}$-115} i & -0.5-0.866025 i & -0.5+0.866025 i & -0.5+0.866025 i & -0.5-0.866025 i & 0. & 0. & 0. \\ 1. & -1.-\text{6.838328748113474$\grave{ }$*${}^{\wedge}$-115} i & -0.5+0.866025 i & -0.5-0.866025 i & 0.5\, +0.866025 i & 0.5\, -0.866025 i & 0. & 0. & 0. \\ 1. & -1.+\text{6.838328748113474$\grave{ }$*${}^{\wedge}$-115} i & -0.5-0.866025 i & -0.5+0.866025 i & 0.5\, -0.866025 i & 0.5\, +0.866025 i & 0. & 0. & 0. \\ \hline \end{array}

The numeric character table is the following

1243657891.1.1.1.1.1.2.514142.514143.320881.1.1.1.1.1.0.5719930.571993−2.672821.1.1.1.1.1.−2.08613−2.086131.351941.−1.1.1.−1.−1.−1.732051.732050.1.−1.1.1.−1.−1.1.73205−1.732050.1.1. +3.108331249142488ˋ*∧-115i−0.5+0.866025i−0.5−0.866025i−0.5−0.866025i−0.5+0.866025i0.0.0.1.1. −3.108331249142488ˋ*∧-115i−0.5−0.866025i−0.5+0.866025i−0.5+0.866025i−0.5−0.866025i0.0.0.1.−1.−6.838328748113474ˋ*∧-115i−0.5+0.866025i−0.5−0.866025i0.5 +0.866025i0.5 −0.866025i0.0.0.1.−1.+6.838328748113474ˋ*∧-115i−0.5−0.866025i−0.5+0.866025i0.5 −0.866025i0.5 +0.866025i0.0.0.\begin{array}{|rrrrrrrrr|} \hline \mathbf{1} & \mathbf{2} & \mathbf{4} & \mathbf{3} & \mathbf{6} & \mathbf{5} & \mathbf{7} & \mathbf{8} & \mathbf{9} \\ \hline 1. & 1. & 1. & 1. & 1. & 1. & 2.51414 & 2.51414 & 3.32088 \\ 1. & 1. & 1. & 1. & 1. & 1. & 0.571993 & 0.571993 & -2.67282 \\ 1. & 1. & 1. & 1. & 1. & 1. & -2.08613 & -2.08613 & 1.35194 \\ 1. & -1. & 1. & 1. & -1. & -1. & -1.73205 & 1.73205 & 0. \\ 1. & -1. & 1. & 1. & -1. & -1. & 1.73205 & -1.73205 & 0. \\ 1. & 1.\, +\text{3.108331249142488$\grave{ }$*${}^{\wedge}$-115} i & -0.5+0.866025 i & -0.5-0.866025 i & -0.5-0.866025 i & -0.5+0.866025 i & 0. & 0. & 0. \\ 1. & 1.\, -\text{3.108331249142488$\grave{ }$*${}^{\wedge}$-115} i & -0.5-0.866025 i & -0.5+0.866025 i & -0.5+0.866025 i & -0.5-0.866025 i & 0. & 0. & 0. \\ 1. & -1.-\text{6.838328748113474$\grave{ }$*${}^{\wedge}$-115} i & -0.5+0.866025 i & -0.5-0.866025 i & 0.5\, +0.866025 i & 0.5\, -0.866025 i & 0. & 0. & 0. \\ 1. & -1.+\text{6.838328748113474$\grave{ }$*${}^{\wedge}$-115} i & -0.5-0.866025 i & -0.5+0.866025 i & 0.5\, -0.866025 i & 0.5\, +0.866025 i & 0. & 0. & 0. \\ \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: