AnyonWiki

\(\text{Ising×\times Ising}:\ \text{FR}^{9,0}_{3}\)

Fusion Rules

12345678921438675934128765943215768958851+4992+36+7667791+23+495+8776693+41+295+885582+3991+46+799996+75+85+86+71+2+3+4\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{8} & \mathbf{6} & \mathbf{7} & \mathbf{5} & \mathbf{9} \\ \mathbf{3} & \mathbf{4} & \mathbf{1} & \mathbf{2} & \mathbf{8} & \mathbf{7} & \mathbf{6} & \mathbf{5} & \mathbf{9} \\ \mathbf{4} & \mathbf{3} & \mathbf{2} & \mathbf{1} & \mathbf{5} & \mathbf{7} & \mathbf{6} & \mathbf{8} & \mathbf{9} \\ \mathbf{5} & \mathbf{8} & \mathbf{8} & \mathbf{5} & \mathbf{1}+\mathbf{4} & \mathbf{9} & \mathbf{9} & \mathbf{2}+\mathbf{3} & \mathbf{6}+\mathbf{7} \\ \mathbf{6} & \mathbf{6} & \mathbf{7} & \mathbf{7} & \mathbf{9} & \mathbf{1}+\mathbf{2} & \mathbf{3}+\mathbf{4} & \mathbf{9} & \mathbf{5}+\mathbf{8} \\ \mathbf{7} & \mathbf{7} & \mathbf{6} & \mathbf{6} & \mathbf{9} & \mathbf{3}+\mathbf{4} & \mathbf{1}+\mathbf{2} & \mathbf{9} & \mathbf{5}+\mathbf{8} \\ \mathbf{8} & \mathbf{5} & \mathbf{5} & \mathbf{8} & \mathbf{2}+\mathbf{3} & \mathbf{9} & \mathbf{9} & \mathbf{1}+\mathbf{4} & \mathbf{6}+\mathbf{7} \\ \mathbf{9} & \mathbf{9} & \mathbf{9} & \mathbf{9} & \mathbf{6}+\mathbf{7} & \mathbf{5}+\mathbf{8} & \mathbf{5}+\mathbf{8} & \mathbf{6}+\mathbf{7} & \mathbf{1}+\mathbf{2}+\mathbf{3}+\mathbf{4} \\ \hline \end{array}

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

{(5 8),(6 7),(2 4)(5 6 8 7),(2 4)(5 7 8 6)}\{(\mathbf{5} \ \mathbf{8}), (\mathbf{6} \ \mathbf{7}), (\mathbf{2} \ \mathbf{4}) (\mathbf{5} \ \mathbf{6} \ \mathbf{8} \ \mathbf{7}), (\mathbf{2} \ \mathbf{4}) (\mathbf{5} \ \mathbf{7} \ \mathbf{8} \ \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}\{\mathbf{1},\mathbf{3}\} Z2: FR12,0\mathbb{Z}_2:\ \text{FR}^{2,0}_{1}
{1,4}\{\mathbf{1},\mathbf{4}\} Z2: FR12,0\mathbb{Z}_2:\ \text{FR}^{2,0}_{1}
{1,2,6}\{\mathbf{1},\mathbf{2},\mathbf{6}\} Ising: FR13,0\text{Ising}:\ \text{FR}^{3,0}_{1}
{1,2,7}\{\mathbf{1},\mathbf{2},\mathbf{7}\} Ising: FR13,0\text{Ising}:\ \text{FR}^{3,0}_{1}
{1,4,5}\{\mathbf{1},\mathbf{4},\mathbf{5}\} Ising: FR13,0\text{Ising}:\ \text{FR}^{3,0}_{1}
{1,4,8}\{\mathbf{1},\mathbf{4},\mathbf{8}\} Ising: FR13,0\text{Ising}:\ \text{FR}^{3,0}_{1}
{1,2,3,4}\{\mathbf{1},\mathbf{2},\mathbf{3},\mathbf{4}\} Z2×Z2: FR14,0\mathbb{Z}_2\times \mathbb{Z}_2:\ \text{FR}^{4,0}_{1}
{1,2,3,4,9}\{\mathbf{1},\mathbf{2},\mathbf{3},\mathbf{4},\mathbf{9}\} Rep(D4): FR15,0\left.\text{Rep(}D_4\right):\ \text{FR}^{5,0}_{1}
{1,2,3,4,5,8}\{\mathbf{1},\mathbf{2},\mathbf{3},\mathbf{4},\mathbf{5},\mathbf{8}\} \(\mathbb{Z}_2\text{×\times Ising}:\ \text{FR}^{6,0}_{1}\)
{1,2,3,4,6,7}\{\mathbf{1},\mathbf{2},\mathbf{3},\mathbf{4},\mathbf{6},\mathbf{7}\} \(\mathbb{Z}_2\text{×\times Ising}:\ \text{FR}^{6,0}_{1}\)

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} 1.414211.41421 2\sqrt{2}
6\mathbf{6} 1.414211.41421 2\sqrt{2}
7\mathbf{7} 1.414211.41421 2\sqrt{2}
8\mathbf{8} 1.414211.41421 2\sqrt{2}
9\mathbf{9} 2.2. 22
DFP2\mathcal{D}_{FP}^2 16.16. 1616

Characters

The symbolic character table is the following

1234867591111222221111−222−2−211112−2−22−21111−2−2−2−221−11−10000011−1−102−2001−1−11200−2011−1−10−22001−1−11−20020\begin{array}{|ccccccccc|} \hline \mathbf{1} & \mathbf{2} & \mathbf{3} & \mathbf{4} & \mathbf{8} & \mathbf{6} & \mathbf{7} & \mathbf{5} & \mathbf{9} \\ \hline 1 & 1 & 1 & 1 & \sqrt{2} & \sqrt{2} & \sqrt{2} & \sqrt{2} & 2 \\ 1 & 1 & 1 & 1 & -\sqrt{2} & \sqrt{2} & \sqrt{2} & -\sqrt{2} & -2 \\ 1 & 1 & 1 & 1 & \sqrt{2} & -\sqrt{2} & -\sqrt{2} & \sqrt{2} & -2 \\ 1 & 1 & 1 & 1 & -\sqrt{2} & -\sqrt{2} & -\sqrt{2} & -\sqrt{2} & 2 \\ 1 & -1 & 1 & -1 & 0 & 0 & 0 & 0 & 0 \\ 1 & 1 & -1 & -1 & 0 & \sqrt{2} & -\sqrt{2} & 0 & 0 \\ 1 & -1 & -1 & 1 & \sqrt{2} & 0 & 0 & -\sqrt{2} & 0 \\ 1 & 1 & -1 & -1 & 0 & -\sqrt{2} & \sqrt{2} & 0 & 0 \\ 1 & -1 & -1 & 1 & -\sqrt{2} & 0 & 0 & \sqrt{2} & 0 \\ \hline \end{array}

The numeric character table is the following

1234867591.0001.0001.0001.0001.4141.4141.4141.4142.0001.0001.0001.0001.000−1.4141.4141.414−1.414−2.0001.0001.0001.0001.0001.414−1.414−1.4141.414−2.0001.0001.0001.0001.000−1.414−1.414−1.414−1.4142.0001.000−1.0001.000−1.000000001.0001.000−1.000−1.00001.414−1.414001.000−1.000−1.0001.0001.41400−1.41401.0001.000−1.000−1.0000−1.4141.414001.000−1.000−1.0001.000−1.414001.4140\begin{array}{|rrrrrrrrr|} \hline \mathbf{1} & \mathbf{2} & \mathbf{3} & \mathbf{4} & \mathbf{8} & \mathbf{6} & \mathbf{7} & \mathbf{5} & \mathbf{9} \\ \hline 1.000 & 1.000 & 1.000 & 1.000 & 1.414 & 1.414 & 1.414 & 1.414 & 2.000 \\ 1.000 & 1.000 & 1.000 & 1.000 & -1.414 & 1.414 & 1.414 & -1.414 & -2.000 \\ 1.000 & 1.000 & 1.000 & 1.000 & 1.414 & -1.414 & -1.414 & 1.414 & -2.000 \\ 1.000 & 1.000 & 1.000 & 1.000 & -1.414 & -1.414 & -1.414 & -1.414 & 2.000 \\ 1.000 & -1.000 & 1.000 & -1.000 & 0 & 0 & 0 & 0 & 0 \\ 1.000 & 1.000 & -1.000 & -1.000 & 0 & 1.414 & -1.414 & 0 & 0 \\ 1.000 & -1.000 & -1.000 & 1.000 & 1.414 & 0 & 0 & -1.414 & 0 \\ 1.000 & 1.000 & -1.000 & -1.000 & 0 & -1.414 & 1.414 & 0 & 0 \\ 1.000 & -1.000 & -1.000 & 1.000 & -1.414 & 0 & 0 & 1.414 & 0 \\ \hline \end{array}

Modular Data

The matching SS-matrices and twist factors are the following

SS-matrix Twist factors
14(11112222211112−2−22−21111−2−2−2−221111−222−2−222−2−202−2002−2−22200−202−2−22−2002022−2−20−22002−22−200000)\frac{1}{4}\left(\begin{array}{ccccccccc} 1 & 1 & 1 & 1 & \sqrt{2} & \sqrt{2} & \sqrt{2} & \sqrt{2} & 2 \\ 1 & 1 & 1 & 1 & \sqrt{2} & -\sqrt{2} & -\sqrt{2} & \sqrt{2} & -2 \\ 1 & 1 & 1 & 1 & -\sqrt{2} & -\sqrt{2} & -\sqrt{2} & -\sqrt{2} & 2 \\ 1 & 1 & 1 & 1 & -\sqrt{2} & \sqrt{2} & \sqrt{2} & -\sqrt{2} & -2 \\ \sqrt{2} & \sqrt{2} & -\sqrt{2} & -\sqrt{2} & 0 & 2 & -2 & 0 & 0 \\ \sqrt{2} & -\sqrt{2} & -\sqrt{2} & \sqrt{2} & 2 & 0 & 0 & -2 & 0 \\ \sqrt{2} & -\sqrt{2} & -\sqrt{2} & \sqrt{2} & -2 & 0 & 0 & 2 & 0 \\ \sqrt{2} & \sqrt{2} & -\sqrt{2} & -\sqrt{2} & 0 & -2 & 2 & 0 & 0 \\ 2 & -2 & 2 & -2 & 0 & 0 & 0 & 0 & 0 \\\end{array}\right) (0,12,0,12,−716,−716,116,116,18)(0,12,0,12,116,116,−716,−716,18)(0,12,0,12,−516,−716,116,316,14)(0,12,0,12,316,116,−716,−516,14)(0,12,0,12,−716,−516,316,116,14)(0,12,0,12,116,316,−516,−716,14)(0,12,0,12,−316,−716,116,516,38)(0,12,0,12,516,116,−716,−316,38)(0,12,0,12,−716,−316,516,116,38)(0,12,0,12,116,516,−316,−716,38)(0,12,0,12,−116,−716,116,716,12)(0,12,0,12,716,116,−716,−116,12)(0,12,0,12,−716,−116,716,116,12)(0,12,0,12,116,716,−116,−716,12)(0,12,0,12,116,−716,116,−716,−38)(0,12,0,12,−716,116,−716,116,−38)(0,12,0,12,316,−716,116,−516,−14)(0,12,0,12,−516,116,−716,316,−14)(0,12,0,12,−716,316,−516,116,−14)(0,12,0,12,116,−516,316,−716,−14)(0,12,0,12,516,−716,116,−316,−18)(0,12,0,12,−316,116,−716,516,−18)(0,12,0,12,−716,516,−316,116,−18)(0,12,0,12,116,−316,516,−716,−18)(0,12,0,12,716,−716,116,−116,0)(0,12,0,12,−116,116,−716,716,0)(0,12,0,12,−716,716,−116,116,0)(0,12,0,12,116,−116,716,−716,0)(0,12,0,12,−516,−516,316,316,38)(0,12,0,12,316,316,−516,−516,38)(0,12,0,12,−316,−516,316,516,12)(0,12,0,12,516,316,−516,−316,12)(0,12,0,12,−516,−316,516,316,12)(0,12,0,12,316,516,−316,−516,12)(0,12,0,12,−116,−516,316,716,−38)(0,12,0,12,716,316,−516,−116,−38)(0,12,0,12,−516,−116,716,316,−38)(0,12,0,12,316,716,−116,−516,−38)(0,12,0,12,316,−516,316,−516,−18)(0,12,0,12,−516,316,−516,316,−18)(0,12,0,12,516,−516,316,−316,0)(0,12,0,12,−316,316,−516,516,0)(0,12,0,12,−516,516,−316,316,0)(0,12,0,12,316,−316,516,−516,0)(0,12,0,12,716,−516,316,−116,18)(0,12,0,12,−116,316,−516,716,18)(0,12,0,12,−516,716,−116,316,18)(0,12,0,12,316,−116,716,−516,18)(0,12,0,12,−316,−316,516,516,−38)(0,12,0,12,516,516,−316,−316,−38)(0,12,0,12,−116,−316,516,716,−14)(0,12,0,12,716,516,−316,−116,−14)(0,12,0,12,−316,−116,716,516,−14)(0,12,0,12,516,716,−116,−316,−14)(0,12,0,12,516,−316,516,−316,18)(0,12,0,12,−316,516,−316,516,18)(0,12,0,12,716,−316,516,−116,14)(0,12,0,12,−116,516,−316,716,14)(0,12,0,12,−316,716,−116,516,14)(0,12,0,12,516,−116,716,−316,14)(0,12,0,12,−116,−116,716,716,−18)(0,12,0,12,716,716,−116,−116,−18)(0,12,0,12,716,−116,716,−116,38)(0,12,0,12,−116,716,−116,716,38)\begin{array}{l}\left(0,\frac{1}{2},0,\frac{1}{2},-\frac{7}{16},-\frac{7}{16},\frac{1}{16},\frac{1}{16},\frac{1}{8}\right) \\\left(0,\frac{1}{2},0,\frac{1}{2},\frac{1}{16},\frac{1}{16},-\frac{7}{16},-\frac{7}{16},\frac{1}{8}\right) \\\left(0,\frac{1}{2},0,\frac{1}{2},-\frac{5}{16},-\frac{7}{16},\frac{1}{16},\frac{3}{16},\frac{1}{4}\right) \\\left(0,\frac{1}{2},0,\frac{1}{2},\frac{3}{16},\frac{1}{16},-\frac{7}{16},-\frac{5}{16},\frac{1}{4}\right) \\\left(0,\frac{1}{2},0,\frac{1}{2},-\frac{7}{16},-\frac{5}{16},\frac{3}{16},\frac{1}{16},\frac{1}{4}\right) \\\left(0,\frac{1}{2},0,\frac{1}{2},\frac{1}{16},\frac{3}{16},-\frac{5}{16},-\frac{7}{16},\frac{1}{4}\right) \\\left(0,\frac{1}{2},0,\frac{1}{2},-\frac{3}{16},-\frac{7}{16},\frac{1}{16},\frac{5}{16},\frac{3}{8}\right) \\\left(0,\frac{1}{2},0,\frac{1}{2},\frac{5}{16},\frac{1}{16},-\frac{7}{16},-\frac{3}{16},\frac{3}{8}\right) \\\left(0,\frac{1}{2},0,\frac{1}{2},-\frac{7}{16},-\frac{3}{16},\frac{5}{16},\frac{1}{16},\frac{3}{8}\right) \\\left(0,\frac{1}{2},0,\frac{1}{2},\frac{1}{16},\frac{5}{16},-\frac{3}{16},-\frac{7}{16},\frac{3}{8}\right) \\\left(0,\frac{1}{2},0,\frac{1}{2},-\frac{1}{16},-\frac{7}{16},\frac{1}{16},\frac{7}{16},\frac{1}{2}\right) \\\left(0,\frac{1}{2},0,\frac{1}{2},\frac{7}{16},\frac{1}{16},-\frac{7}{16},-\frac{1}{16},\frac{1}{2}\right) \\\left(0,\frac{1}{2},0,\frac{1}{2},-\frac{7}{16},-\frac{1}{16},\frac{7}{16},\frac{1}{16},\frac{1}{2}\right) \\\left(0,\frac{1}{2},0,\frac{1}{2},\frac{1}{16},\frac{7}{16},-\frac{1}{16},-\frac{7}{16},\frac{1}{2}\right) \\\left(0,\frac{1}{2},0,\frac{1}{2},\frac{1}{16},-\frac{7}{16},\frac{1}{16},-\frac{7}{16},-\frac{3}{8}\right) \\\left(0,\frac{1}{2},0,\frac{1}{2},-\frac{7}{16},\frac{1}{16},-\frac{7}{16},\frac{1}{16},-\frac{3}{8}\right) \\\left(0,\frac{1}{2},0,\frac{1}{2},\frac{3}{16},-\frac{7}{16},\frac{1}{16},-\frac{5}{16},-\frac{1}{4}\right) \\\left(0,\frac{1}{2},0,\frac{1}{2},-\frac{5}{16},\frac{1}{16},-\frac{7}{16},\frac{3}{16},-\frac{1}{4}\right) \\\left(0,\frac{1}{2},0,\frac{1}{2},-\frac{7}{16},\frac{3}{16},-\frac{5}{16},\frac{1}{16},-\frac{1}{4}\right) \\\left(0,\frac{1}{2},0,\frac{1}{2},\frac{1}{16},-\frac{5}{16},\frac{3}{16},-\frac{7}{16},-\frac{1}{4}\right) \\\left(0,\frac{1}{2},0,\frac{1}{2},\frac{5}{16},-\frac{7}{16},\frac{1}{16},-\frac{3}{16},-\frac{1}{8}\right) \\\left(0,\frac{1}{2},0,\frac{1}{2},-\frac{3}{16},\frac{1}{16},-\frac{7}{16},\frac{5}{16},-\frac{1}{8}\right) \\\left(0,\frac{1}{2},0,\frac{1}{2},-\frac{7}{16},\frac{5}{16},-\frac{3}{16},\frac{1}{16},-\frac{1}{8}\right) \\\left(0,\frac{1}{2},0,\frac{1}{2},\frac{1}{16},-\frac{3}{16},\frac{5}{16},-\frac{7}{16},-\frac{1}{8}\right) \\\left(0,\frac{1}{2},0,\frac{1}{2},\frac{7}{16},-\frac{7}{16},\frac{1}{16},-\frac{1}{16},0\right) \\\left(0,\frac{1}{2},0,\frac{1}{2},-\frac{1}{16},\frac{1}{16},-\frac{7}{16},\frac{7}{16},0\right) \\\left(0,\frac{1}{2},0,\frac{1}{2},-\frac{7}{16},\frac{7}{16},-\frac{1}{16},\frac{1}{16},0\right) \\\left(0,\frac{1}{2},0,\frac{1}{2},\frac{1}{16},-\frac{1}{16},\frac{7}{16},-\frac{7}{16},0\right) \\\left(0,\frac{1}{2},0,\frac{1}{2},-\frac{5}{16},-\frac{5}{16},\frac{3}{16},\frac{3}{16},\frac{3}{8}\right) \\\left(0,\frac{1}{2},0,\frac{1}{2},\frac{3}{16},\frac{3}{16},-\frac{5}{16},-\frac{5}{16},\frac{3}{8}\right) \\\left(0,\frac{1}{2},0,\frac{1}{2},-\frac{3}{16},-\frac{5}{16},\frac{3}{16},\frac{5}{16},\frac{1}{2}\right) \\\left(0,\frac{1}{2},0,\frac{1}{2},\frac{5}{16},\frac{3}{16},-\frac{5}{16},-\frac{3}{16},\frac{1}{2}\right) \\\left(0,\frac{1}{2},0,\frac{1}{2},-\frac{5}{16},-\frac{3}{16},\frac{5}{16},\frac{3}{16},\frac{1}{2}\right) \\\left(0,\frac{1}{2},0,\frac{1}{2},\frac{3}{16},\frac{5}{16},-\frac{3}{16},-\frac{5}{16},\frac{1}{2}\right) \\\left(0,\frac{1}{2},0,\frac{1}{2},-\frac{1}{16},-\frac{5}{16},\frac{3}{16},\frac{7}{16},-\frac{3}{8}\right) \\\left(0,\frac{1}{2},0,\frac{1}{2},\frac{7}{16},\frac{3}{16},-\frac{5}{16},-\frac{1}{16},-\frac{3}{8}\right) \\\left(0,\frac{1}{2},0,\frac{1}{2},-\frac{5}{16},-\frac{1}{16},\frac{7}{16},\frac{3}{16},-\frac{3}{8}\right) \\\left(0,\frac{1}{2},0,\frac{1}{2},\frac{3}{16},\frac{7}{16},-\frac{1}{16},-\frac{5}{16},-\frac{3}{8}\right) \\\left(0,\frac{1}{2},0,\frac{1}{2},\frac{3}{16},-\frac{5}{16},\frac{3}{16},-\frac{5}{16},-\frac{1}{8}\right) \\\left(0,\frac{1}{2},0,\frac{1}{2},-\frac{5}{16},\frac{3}{16},-\frac{5}{16},\frac{3}{16},-\frac{1}{8}\right) \\\left(0,\frac{1}{2},0,\frac{1}{2},\frac{5}{16},-\frac{5}{16},\frac{3}{16},-\frac{3}{16},0\right) \\\left(0,\frac{1}{2},0,\frac{1}{2},-\frac{3}{16},\frac{3}{16},-\frac{5}{16},\frac{5}{16},0\right) \\\left(0,\frac{1}{2},0,\frac{1}{2},-\frac{5}{16},\frac{5}{16},-\frac{3}{16},\frac{3}{16},0\right) \\\left(0,\frac{1}{2},0,\frac{1}{2},\frac{3}{16},-\frac{3}{16},\frac{5}{16},-\frac{5}{16},0\right) \\\left(0,\frac{1}{2},0,\frac{1}{2},\frac{7}{16},-\frac{5}{16},\frac{3}{16},-\frac{1}{16},\frac{1}{8}\right) \\\left(0,\frac{1}{2},0,\frac{1}{2},-\frac{1}{16},\frac{3}{16},-\frac{5}{16},\frac{7}{16},\frac{1}{8}\right) \\\left(0,\frac{1}{2},0,\frac{1}{2},-\frac{5}{16},\frac{7}{16},-\frac{1}{16},\frac{3}{16},\frac{1}{8}\right) \\\left(0,\frac{1}{2},0,\frac{1}{2},\frac{3}{16},-\frac{1}{16},\frac{7}{16},-\frac{5}{16},\frac{1}{8}\right) \\\left(0,\frac{1}{2},0,\frac{1}{2},-\frac{3}{16},-\frac{3}{16},\frac{5}{16},\frac{5}{16},-\frac{3}{8}\right) \\\left(0,\frac{1}{2},0,\frac{1}{2},\frac{5}{16},\frac{5}{16},-\frac{3}{16},-\frac{3}{16},-\frac{3}{8}\right) \\\left(0,\frac{1}{2},0,\frac{1}{2},-\frac{1}{16},-\frac{3}{16},\frac{5}{16},\frac{7}{16},-\frac{1}{4}\right) \\\left(0,\frac{1}{2},0,\frac{1}{2},\frac{7}{16},\frac{5}{16},-\frac{3}{16},-\frac{1}{16},-\frac{1}{4}\right) \\\left(0,\frac{1}{2},0,\frac{1}{2},-\frac{3}{16},-\frac{1}{16},\frac{7}{16},\frac{5}{16},-\frac{1}{4}\right) \\\left(0,\frac{1}{2},0,\frac{1}{2},\frac{5}{16},\frac{7}{16},-\frac{1}{16},-\frac{3}{16},-\frac{1}{4}\right) \\\left(0,\frac{1}{2},0,\frac{1}{2},\frac{5}{16},-\frac{3}{16},\frac{5}{16},-\frac{3}{16},\frac{1}{8}\right) \\\left(0,\frac{1}{2},0,\frac{1}{2},-\frac{3}{16},\frac{5}{16},-\frac{3}{16},\frac{5}{16},\frac{1}{8}\right) \\\left(0,\frac{1}{2},0,\frac{1}{2},\frac{7}{16},-\frac{3}{16},\frac{5}{16},-\frac{1}{16},\frac{1}{4}\right) \\\left(0,\frac{1}{2},0,\frac{1}{2},-\frac{1}{16},\frac{5}{16},-\frac{3}{16},\frac{7}{16},\frac{1}{4}\right) \\\left(0,\frac{1}{2},0,\frac{1}{2},-\frac{3}{16},\frac{7}{16},-\frac{1}{16},\frac{5}{16},\frac{1}{4}\right) \\\left(0,\frac{1}{2},0,\frac{1}{2},\frac{5}{16},-\frac{1}{16},\frac{7}{16},-\frac{3}{16},\frac{1}{4}\right) \\\left(0,\frac{1}{2},0,\frac{1}{2},-\frac{1}{16},-\frac{1}{16},\frac{7}{16},\frac{7}{16},-\frac{1}{8}\right) \\\left(0,\frac{1}{2},0,\frac{1}{2},\frac{7}{16},\frac{7}{16},-\frac{1}{16},-\frac{1}{16},-\frac{1}{8}\right) \\\left(0,\frac{1}{2},0,\frac{1}{2},\frac{7}{16},-\frac{1}{16},\frac{7}{16},-\frac{1}{16},\frac{3}{8}\right) \\\left(0,\frac{1}{2},0,\frac{1}{2},-\frac{1}{16},\frac{7}{16},-\frac{1}{16},\frac{7}{16},\frac{3}{8}\right)\end{array}

Adjoint Subring

Particles 1,2,3,4\mathbf{1}, \mathbf{2}, \mathbf{3}, \mathbf{4}, form the adjoint subring Z2×Z2: FR14,0\mathbb{Z}_2\times \mathbb{Z}_2:\ \text{FR}^{4,0}_{1} .

The upper central series is the following: \(\text{Ising×\times Ising} \underset{ \mathbf{1}, \mathbf{2}, \mathbf{3}, \mathbf{4} }{\supset} \mathbb{Z}_2\times \mathbb{Z}_2 \underset{ \mathbf{1} }{\supset} \text{Trivial}\)

Universal grading

Each particle can be graded as follows: deg(1)=1′,deg(2)=1′,deg(3)=1′,deg(4)=1′,deg(5)=2′,deg(6)=3′,deg(7)=3′,deg(8)=2′,deg(9)=4′\text{deg}(\mathbf{1}) = \mathbf{1}', \text{deg}(\mathbf{2}) = \mathbf{1}', \text{deg}(\mathbf{3}) = \mathbf{1}', \text{deg}(\mathbf{4}) = \mathbf{1}', \text{deg}(\mathbf{5}) = \mathbf{2}', \text{deg}(\mathbf{6}) = \mathbf{3}', \text{deg}(\mathbf{7}) = \mathbf{3}', \text{deg}(\mathbf{8}) = \mathbf{2}', \text{deg}(\mathbf{9}) = \mathbf{4}', where the degrees form the group Z2×Z2\mathbb{Z}_2\times \mathbb{Z}_2 with multiplication table:

1′2′3′4′2′1′4′3′3′4′1′2′4′3′2′1′\begin{array}{|llll|} \hline \mathbf{1}' & \mathbf{2}' & \mathbf{3}' & \mathbf{4}' \\ \mathbf{2}' & \mathbf{1}' & \mathbf{4}' & \mathbf{3}' \\ \mathbf{3}' & \mathbf{4}' & \mathbf{1}' & \mathbf{2}' \\ \mathbf{4}' & \mathbf{3}' & \mathbf{2}' & \mathbf{1}' \\ \hline \end{array}

Categorifications

Data

Download links for numeric data: