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FR99,4\text{FR}^{9,4}_{9}

Fusion Rules

12345678921435678934215896743125896755551+2+3+47+96+87+96+866997+81+2+65+85+73+4+977886+95+91+2+73+4+85+688776+93+4+85+65+91+2+799667+85+73+4+91+2+65+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{4} & \mathbf{3} & \mathbf{5} & \mathbf{6} & \mathbf{7} & \mathbf{8} & \mathbf{9} \\ \mathbf{3} & \mathbf{4} & \mathbf{2} & \mathbf{1} & \mathbf{5} & \mathbf{8} & \mathbf{9} & \mathbf{6} & \mathbf{7} \\ \mathbf{4} & \mathbf{3} & \mathbf{1} & \mathbf{2} & \mathbf{5} & \mathbf{8} & \mathbf{9} & \mathbf{6} & \mathbf{7} \\ \mathbf{5} & \mathbf{5} & \mathbf{5} & \mathbf{5} & \mathbf{1}+\mathbf{2}+\mathbf{3}+\mathbf{4} & \mathbf{7}+\mathbf{9} & \mathbf{6}+\mathbf{8} & \mathbf{7}+\mathbf{9} & \mathbf{6}+\mathbf{8} \\ \mathbf{6} & \mathbf{6} & \mathbf{9} & \mathbf{9} & \mathbf{7}+\mathbf{8} & \mathbf{1}+\mathbf{2}+\mathbf{6} & \mathbf{5}+\mathbf{8} & \mathbf{5}+\mathbf{7} & \mathbf{3}+\mathbf{4}+\mathbf{9} \\ \mathbf{7} & \mathbf{7} & \mathbf{8} & \mathbf{8} & \mathbf{6}+\mathbf{9} & \mathbf{5}+\mathbf{9} & \mathbf{1}+\mathbf{2}+\mathbf{7} & \mathbf{3}+\mathbf{4}+\mathbf{8} & \mathbf{5}+\mathbf{6} \\ \mathbf{8} & \mathbf{8} & \mathbf{7} & \mathbf{7} & \mathbf{6}+\mathbf{9} & \mathbf{3}+\mathbf{4}+\mathbf{8} & \mathbf{5}+\mathbf{6} & \mathbf{5}+\mathbf{9} & \mathbf{1}+\mathbf{2}+\mathbf{7} \\ \mathbf{9} & \mathbf{9} & \mathbf{6} & \mathbf{6} & \mathbf{7}+\mathbf{8} & \mathbf{5}+\mathbf{7} & \mathbf{3}+\mathbf{4}+\mathbf{9} & \mathbf{1}+\mathbf{2}+\mathbf{6} & \mathbf{5}+\mathbf{8} \\ \hline \end{array}

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

{(3 4),(6 7)(8 9)}\{(\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}
{1,2,6}\{\mathbf{1},\mathbf{2},\mathbf{6}\} Rep(D3): FR23,0\left.\text{Rep(}D_3\right):\ \text{FR}^{3,0}_{2}
{1,2,7}\{\mathbf{1},\mathbf{2},\mathbf{7}\} Rep(D3): FR23,0\left.\text{Rep(}D_3\right):\ \text{FR}^{3,0}_{2}
{1,2,3,4}\{\mathbf{1},\mathbf{2},\mathbf{3},\mathbf{4}\} Z4: FR14,2\mathbb{Z}_4:\ \text{FR}^{4,2}_{1}
{1,2,3,4,5}\{\mathbf{1},\mathbf{2},\mathbf{3},\mathbf{4},\mathbf{5}\} TY(Z4): FR15,2\left.\text{TY(}\mathbb{Z}_4\right):\ \text{FR}^{5,2}_{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} 2.2. 22
6\mathbf{6} 2.2. 22
7\mathbf{7} 2.2. 22
8\mathbf{8} 2.2. 22
9\mathbf{9} 2.2. 22
DFP2\mathcal{D}_{FP}^2 24.24. 2424

Characters

The symbolic character table is the following

13425897611112222211112−1−1−1−11−1−11011−1−11i−i−1000001−ii−100000212122−52−34−34−1203120\begin{array}{|ccccccccc|} \hline \mathbf{1} & \mathbf{3} & \mathbf{4} & \mathbf{2} & \mathbf{5} & \mathbf{8} & \mathbf{9} & \mathbf{7} & \mathbf{6} \\ \hline 1 & 1 & 1 & 1 & 2 & 2 & 2 & 2 & 2 \\ 1 & 1 & 1 & 1 & 2 & -1 & -1 & -1 & -1 \\ 1 & -1 & -1 & 1 & 0 & 1 & 1 & -1 & -1 \\ 1 & i & -i & -1 & 0 & 0 & 0 & 0 & 0 \\ 1 & -i & i & -1 & 0 & 0 & 0 & 0 & 0 \\ 2 & \frac{1}{2} & \frac{1}{2} & 2 & -\frac{5}{2} & -\frac{3}{4} & -\frac{3}{4} & -\frac{1}{20} & \frac{31}{20} \\ \hline \end{array}

The numeric character table is the following

1342589761.0001.0001.0001.0002.0002.0002.0002.0002.0001.0001.0001.0001.0002.000−1.000−1.000−1.000−1.0001.000−1.000−1.0001.00001.0001.000−1.000−1.0001.0001.000i−1.000i−1.000000001.000−1.000i1.000i−1.000000002.0000.50000.50002.000−2.500−0.7500−0.7500−0.050001.550\begin{array}{|rrrrrrrrr|} \hline \mathbf{1} & \mathbf{3} & \mathbf{4} & \mathbf{2} & \mathbf{5} & \mathbf{8} & \mathbf{9} & \mathbf{7} & \mathbf{6} \\ \hline 1.000 & 1.000 & 1.000 & 1.000 & 2.000 & 2.000 & 2.000 & 2.000 & 2.000 \\ 1.000 & 1.000 & 1.000 & 1.000 & 2.000 & -1.000 & -1.000 & -1.000 & -1.000 \\ 1.000 & -1.000 & -1.000 & 1.000 & 0 & 1.000 & 1.000 & -1.000 & -1.000 \\ 1.000 & 1.000 i & -1.000 i & -1.000 & 0 & 0 & 0 & 0 & 0 \\ 1.000 & -1.000 i & 1.000 i & -1.000 & 0 & 0 & 0 & 0 & 0 \\ 2.000 & 0.5000 & 0.5000 & 2.000 & -2.500 & -0.7500 & -0.7500 & -0.05000 & 1.550 \\ \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

Data

Download links for numeric data: