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\(\text{Fib×\times Potts}:\ \text{FR}^{8,4}_{9}\)

Fusion Rules

1234567823164578312564784651+43+52+687+85463+52+61+487+86542+61+43+587+87778881+2+34+5+68887+87+87+84+5+61+2+3+4+5+6\begin{array}{|llllllll|} \hline \mathbf{1} & \mathbf{2} & \mathbf{3} & \mathbf{4} & \mathbf{5} & \mathbf{6} & \mathbf{7} & \mathbf{8} \\ \mathbf{2} & \mathbf{3} & \mathbf{1} & \mathbf{6} & \mathbf{4} & \mathbf{5} & \mathbf{7} & \mathbf{8} \\ \mathbf{3} & \mathbf{1} & \mathbf{2} & \mathbf{5} & \mathbf{6} & \mathbf{4} & \mathbf{7} & \mathbf{8} \\ \mathbf{4} & \mathbf{6} & \mathbf{5} & \mathbf{1}+\mathbf{4} & \mathbf{3}+\mathbf{5} & \mathbf{2}+\mathbf{6} & \mathbf{8} & \mathbf{7}+\mathbf{8} \\ \mathbf{5} & \mathbf{4} & \mathbf{6} & \mathbf{3}+\mathbf{5} & \mathbf{2}+\mathbf{6} & \mathbf{1}+\mathbf{4} & \mathbf{8} & \mathbf{7}+\mathbf{8} \\ \mathbf{6} & \mathbf{5} & \mathbf{4} & \mathbf{2}+\mathbf{6} & \mathbf{1}+\mathbf{4} & \mathbf{3}+\mathbf{5} & \mathbf{8} & \mathbf{7}+\mathbf{8} \\ \mathbf{7} & \mathbf{7} & \mathbf{7} & \mathbf{8} & \mathbf{8} & \mathbf{8} & \mathbf{1}+\mathbf{2}+\mathbf{3} & \mathbf{4}+\mathbf{5}+\mathbf{6} \\ \mathbf{8} & \mathbf{8} & \mathbf{8} & \mathbf{7}+\mathbf{8} & \mathbf{7}+\mathbf{8} & \mathbf{7}+\mathbf{8} & \mathbf{4}+\mathbf{5}+\mathbf{6} & \mathbf{1}+\mathbf{2}+\mathbf{3}+\mathbf{4}+\mathbf{5}+\mathbf{6} \\ \hline \end{array}

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

{(2 3)(5 6)}\{(\mathbf{2} \ \mathbf{3}) (\mathbf{5} \ \mathbf{6})\}

The following elements form non-trivial sub fusion rings

Elements SubRing
{1,4}\{\mathbf{1},\mathbf{4}\} Fib: FR22,0\text{Fib}:\ \text{FR}^{2,0}_{2}
{1,2,3}\{\mathbf{1},\mathbf{2},\mathbf{3}\} Z3: FR13,2\mathbb{Z}_3:\ \text{FR}^{3,2}_{1}
{1,2,3,7}\{\mathbf{1},\mathbf{2},\mathbf{3},\mathbf{7}\} Potts: FR24,2\text{Potts}:\ \text{FR}^{4,2}_{2}
{1,2,3,4,5,6}\{\mathbf{1},\mathbf{2},\mathbf{3},\mathbf{4},\mathbf{5},\mathbf{6}\} \(\text{Fib×\times }\mathbb{Z}_3:\ \text{FR}^{6,4}_{5}\)

Frobenius-Perron 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} 1.618031.61803 12(1+5)\frac{1}{2} \left(1+\sqrt{5}\right)
5\mathbf{5} 1.618031.61803 12(1+5)\frac{1}{2} \left(1+\sqrt{5}\right)
6\mathbf{6} 1.618031.61803 12(1+5)\frac{1}{2} \left(1+\sqrt{5}\right)
7\mathbf{7} 1.732051.73205 3\sqrt{3}
8\mathbf{8} 2.802522.80252 32(3+5)\sqrt{\frac{3}{2} \left(3+\sqrt{5}\right)}
DFP2\mathcal{D}_{FP}^2 21.708221.7082 6+34(1+5)2+32(3+5)6+\frac{3}{4} \left(1+\sqrt{5}\right)^2+\frac{3}{2} \left(3+\sqrt{5}\right)

Characters

The symbolic character table is the following

1236547811112(1+5)12(1+5)12(1+5)3Root[x4−9x2+9,4]11112(1+5)12(1+5)12(1+5)−3Root[x4−9x2+9,1]11112(1−5)12(1−5)12(1−5)3Root[x4−9x2+9,2]11112(1−5)12(1−5)12(1−5)−3Root[x4−9x2+9,3]112(−1−i3)12(−1+i3)Root[x4+x3+2x2−x+1,4]Root[x4+x3+2x2−x+1,3]12(1−5)00112(−1+i3)12(−1−i3)Root[x4+x3+2x2−x+1,3]Root[x4+x3+2x2−x+1,4]12(1−5)00112(−1+i3)12(−1−i3)Root[x4+x3+2x2−x+1,2]Root[x4+x3+2x2−x+1,1]12(1+5)00112(−1−i3)12(−1+i3)Root[x4+x3+2x2−x+1,1]Root[x4+x3+2x2−x+1,2]12(1+5)00\begin{array}{|cccccccc|} \hline \mathbf{1} & \mathbf{2} & \mathbf{3} & \mathbf{6} & \mathbf{5} & \mathbf{4} & \mathbf{7} & \mathbf{8} \\ \hline 1 & 1 & 1 & \frac{1}{2} \left(1+\sqrt{5}\right) & \frac{1}{2} \left(1+\sqrt{5}\right) & \frac{1}{2} \left(1+\sqrt{5}\right) & \sqrt{3} & \text{Root}\left[x^4-9 x^2+9,4\right] \\ 1 & 1 & 1 & \frac{1}{2} \left(1+\sqrt{5}\right) & \frac{1}{2} \left(1+\sqrt{5}\right) & \frac{1}{2} \left(1+\sqrt{5}\right) & -\sqrt{3} & \text{Root}\left[x^4-9 x^2+9,1\right] \\ 1 & 1 & 1 & \frac{1}{2} \left(1-\sqrt{5}\right) & \frac{1}{2} \left(1-\sqrt{5}\right) & \frac{1}{2} \left(1-\sqrt{5}\right) & \sqrt{3} & \text{Root}\left[x^4-9 x^2+9,2\right] \\ 1 & 1 & 1 & \frac{1}{2} \left(1-\sqrt{5}\right) & \frac{1}{2} \left(1-\sqrt{5}\right) & \frac{1}{2} \left(1-\sqrt{5}\right) & -\sqrt{3} & \text{Root}\left[x^4-9 x^2+9,3\right] \\ 1 & \frac{1}{2} \left(-1-i \sqrt{3}\right) & \frac{1}{2} \left(-1+i \sqrt{3}\right) & \text{Root}\left[x^4+x^3+2 x^2-x+1,4\right] & \text{Root}\left[x^4+x^3+2 x^2-x+1,3\right] & \frac{1}{2} \left(1-\sqrt{5}\right) & 0 & 0 \\ 1 & \frac{1}{2} \left(-1+i \sqrt{3}\right) & \frac{1}{2} \left(-1-i \sqrt{3}\right) & \text{Root}\left[x^4+x^3+2 x^2-x+1,3\right] & \text{Root}\left[x^4+x^3+2 x^2-x+1,4\right] & \frac{1}{2} \left(1-\sqrt{5}\right) & 0 & 0 \\ 1 & \frac{1}{2} \left(-1+i \sqrt{3}\right) & \frac{1}{2} \left(-1-i \sqrt{3}\right) & \text{Root}\left[x^4+x^3+2 x^2-x+1,2\right] & \text{Root}\left[x^4+x^3+2 x^2-x+1,1\right] & \frac{1}{2} \left(1+\sqrt{5}\right) & 0 & 0 \\ 1 & \frac{1}{2} \left(-1-i \sqrt{3}\right) & \frac{1}{2} \left(-1+i \sqrt{3}\right) & \text{Root}\left[x^4+x^3+2 x^2-x+1,1\right] & \text{Root}\left[x^4+x^3+2 x^2-x+1,2\right] & \frac{1}{2} \left(1+\sqrt{5}\right) & 0 & 0 \\ \hline \end{array}

The numeric character table is the following

123654781.0001.0001.0001.6181.6181.6181.7322.8031.0001.0001.0001.6181.6181.618−1.732−2.8031.0001.0001.000−0.6180−0.6180−0.61801.732−1.0701.0001.0001.000−0.6180−0.6180−0.6180−1.7321.0701.000−0.5000−0.8660i−0.5000+0.8660i0.3090+0.5352i0.3090−0.5352i−0.6180001.000−0.5000+0.8660i−0.5000−0.8660i0.3090−0.5352i0.3090+0.5352i−0.6180001.000−0.5000+0.8660i−0.5000−0.8660i−0.809+1.401i−0.809−1.401i1.618001.000−0.5000−0.8660i−0.5000+0.8660i−0.809−1.401i−0.809+1.401i1.61800\begin{array}{|rrrrrrrr|} \hline \mathbf{1} & \mathbf{2} & \mathbf{3} & \mathbf{6} & \mathbf{5} & \mathbf{4} & \mathbf{7} & \mathbf{8} \\ \hline 1.000 & 1.000 & 1.000 & 1.618 & 1.618 & 1.618 & 1.732 & 2.803 \\ 1.000 & 1.000 & 1.000 & 1.618 & 1.618 & 1.618 & -1.732 & -2.803 \\ 1.000 & 1.000 & 1.000 & -0.6180 & -0.6180 & -0.6180 & 1.732 & -1.070 \\ 1.000 & 1.000 & 1.000 & -0.6180 & -0.6180 & -0.6180 & -1.732 & 1.070 \\ 1.000 & -0.5000-0.8660 i & -0.5000+0.8660 i & 0.3090+0.5352 i & 0.3090-0.5352 i & -0.6180 & 0 & 0 \\ 1.000 & -0.5000+0.8660 i & -0.5000-0.8660 i & 0.3090-0.5352 i & 0.3090+0.5352 i & -0.6180 & 0 & 0 \\ 1.000 & -0.5000+0.8660 i & -0.5000-0.8660 i & -0.809+1.401 i & -0.809-1.401 i & 1.618 & 0 & 0 \\ 1.000 & -0.5000-0.8660 i & -0.5000+0.8660 i & -0.809-1.401 i & -0.809+1.401 i & 1.618 & 0 & 0 \\ \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

Elements 1,2,3,4,5,6\mathbf{1}, \mathbf{2}, \mathbf{3}, \mathbf{4}, \mathbf{5}, \mathbf{6}, form the adjoint subring \(\text{Fib×\times }\mathbb{Z}_3:\ \text{FR}^{6,4}_{5}\) .

The upper central series is the following: \(\text{Fib×\times Potts} \underset{ \mathbf{1}, \mathbf{2}, \mathbf{3}, \mathbf{4}, \mathbf{5}, \mathbf{6} }{\supset} \text{Fib×\times }\mathbb{Z}_3 \underset{ \mathbf{1}, \mathbf{2} }{\supset} \text{Fib}\)

Universal grading

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

1′2′2′1′\begin{array}{|ll|} \hline \mathbf{1}' & \mathbf{2}' \\ \mathbf{2}' & \mathbf{1}' \\ \hline \end{array}

Categorifications

Data

Download links for numeric data: