FR 11 8 , 2 \text{FR}^{8,2}_{11} FR 1 1 8 , 2 Fusion Rules
1 2 3 4 5 6 7 8 2 1 5 6 3 4 8 7 3 6 1 + 3 8 7 2 + 6 5 + 7 4 + 8 4 5 8 1 + 4 2 + 5 7 6 + 7 3 + 8 5 4 2 + 5 7 8 1 + 4 3 + 8 6 + 7 6 3 7 2 + 6 1 + 3 8 4 + 8 5 + 7 7 8 6 + 7 5 + 7 4 + 8 3 + 8 1 + 3 + 4 + 8 2 + 5 + 6 + 7 8 7 4 + 8 3 + 8 6 + 7 5 + 7 2 + 5 + 6 + 7 1 + 3 + 4 + 8 \begin{array}{|llllllll|}
\hline
\mathbf{1} & \mathbf{2} & \mathbf{3} & \mathbf{4} & \mathbf{5} & \mathbf{6} & \mathbf{7} & \mathbf{8} \\
\mathbf{2} & \mathbf{1} & \mathbf{5} & \mathbf{6} & \mathbf{3} & \mathbf{4} & \mathbf{8} & \mathbf{7} \\
\mathbf{3} & \mathbf{6} & \mathbf{1}+\mathbf{3} & \mathbf{8} & \mathbf{7} & \mathbf{2}+\mathbf{6} & \mathbf{5}+\mathbf{7} & \mathbf{4}+\mathbf{8} \\
\mathbf{4} & \mathbf{5} & \mathbf{8} & \mathbf{1}+\mathbf{4} & \mathbf{2}+\mathbf{5} & \mathbf{7} & \mathbf{6}+\mathbf{7} & \mathbf{3}+\mathbf{8} \\
\mathbf{5} & \mathbf{4} & \mathbf{2}+\mathbf{5} & \mathbf{7} & \mathbf{8} & \mathbf{1}+\mathbf{4} & \mathbf{3}+\mathbf{8} & \mathbf{6}+\mathbf{7} \\
\mathbf{6} & \mathbf{3} & \mathbf{7} & \mathbf{2}+\mathbf{6} & \mathbf{1}+\mathbf{3} & \mathbf{8} & \mathbf{4}+\mathbf{8} & \mathbf{5}+\mathbf{7} \\
\mathbf{7} & \mathbf{8} & \mathbf{6}+\mathbf{7} & \mathbf{5}+\mathbf{7} & \mathbf{4}+\mathbf{8} & \mathbf{3}+\mathbf{8} & \mathbf{1}+\mathbf{3}+\mathbf{4}+\mathbf{8} & \mathbf{2}+\mathbf{5}+\mathbf{6}+\mathbf{7} \\
\mathbf{8} & \mathbf{7} & \mathbf{4}+\mathbf{8} & \mathbf{3}+\mathbf{8} & \mathbf{6}+\mathbf{7} & \mathbf{5}+\mathbf{7} & \mathbf{2}+\mathbf{5}+\mathbf{6}+\mathbf{7} & \mathbf{1}+\mathbf{3}+\mathbf{4}+\mathbf{8} \\
\hline
\end{array} 1 2 3 4 5 6 7 8 2 1 6 5 4 3 8 7 3 5 1 + 3 8 2 + 5 7 6 + 7 4 + 8 4 6 8 1 + 4 7 2 + 6 5 + 7 3 + 8 5 3 7 2 + 5 8 1 + 3 4 + 8 6 + 7 6 4 2 + 6 7 1 + 4 8 3 + 8 5 + 7 7 8 5 + 7 6 + 7 3 + 8 4 + 8 1 + 3 + 4 + 8 2 + 5 + 6 + 7 8 7 4 + 8 3 + 8 6 + 7 5 + 7 2 + 5 + 6 + 7 1 + 3 + 4 + 8 The fusion rules are invariant under the group generated by the following permutations:
{ ( 3 4 ) ( 5 6 ) } \{(\mathbf{3} \ \mathbf{4}) (\mathbf{5} \ \mathbf{6})\} { ( 3 4 ) ( 5 6 ) } The following elements form non-trivial sub fusion rings
Elements
SubRing
{ 1 , 2 } \{\mathbf{1},\mathbf{2}\} { 1 , 2 } Z 2 : FR 1 2 , 0 \mathbb{Z}_2:\ \text{FR}^{2,0}_{1} Z 2 : FR 1 2 , 0
{ 1 , 3 } \{\mathbf{1},\mathbf{3}\} { 1 , 3 } Fib : FR 2 2 , 0 \text{Fib}:\ \text{FR}^{2,0}_{2} Fib : FR 2 2 , 0
{ 1 , 4 } \{\mathbf{1},\mathbf{4}\} { 1 , 4 } Fib : FR 2 2 , 0 \text{Fib}:\ \text{FR}^{2,0}_{2} Fib : FR 2 2 , 0
{ 1 , 3 , 4 , 8 } \{\mathbf{1},\mathbf{3},\mathbf{4},\mathbf{8}\} { 1 , 3 , 4 , 8 } \(\text{Fib× \times ×
Frobenius-Perron Dimensions
Particle
Numeric
Symbolic
1 \mathbf{1} 1 1. 1. 1 . 1 1 1
2 \mathbf{2} 2 1. 1. 1 . 1 1 1
3 \mathbf{3} 3 1.61803 1.61803 1 . 6 1 8 0 3 1 2 ( 1 + 5 ) \frac{1}{2} \left(1+\sqrt{5}\right) 2 1 ( 1 + 5 )
4 \mathbf{4} 4 1.61803 1.61803 1 . 6 1 8 0 3 1 2 ( 1 + 5 ) \frac{1}{2} \left(1+\sqrt{5}\right) 2 1 ( 1 + 5 )
5 \mathbf{5} 5 1.61803 1.61803 1 . 6 1 8 0 3 1 2 ( 1 + 5 ) \frac{1}{2} \left(1+\sqrt{5}\right) 2 1 ( 1 + 5 )
6 \mathbf{6} 6 1.61803 1.61803 1 . 6 1 8 0 3 1 2 ( 1 + 5 ) \frac{1}{2} \left(1+\sqrt{5}\right) 2 1 ( 1 + 5 )
7 \mathbf{7} 7 2.61803 2.61803 2 . 6 1 8 0 3 1 2 ( 3 + 5 ) \frac{1}{2} \left(3+\sqrt{5}\right) 2 1 ( 3 + 5 )
8 \mathbf{8} 8 2.61803 2.61803 2 . 6 1 8 0 3 1 2 ( 3 + 5 ) \frac{1}{2} \left(3+\sqrt{5}\right) 2 1 ( 3 + 5 )
D F P 2 \mathcal{D}_{FP}^2 D F P 2 26.1803 26.1803 2 6 . 1 8 0 3 2 + ( 1 + 5 ) 2 + 1 2 ( 3 + 5 ) 2 2+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2 2 + ( 1 + 5 ) 2 + 2 1 ( 3 + 5 ) 2
Characters
The symbolic character table is the following
1 2 5 6 4 3 8 7 1 1 1 2 ( 1 + 5 ) 1 2 ( 1 + 5 ) 1 2 ( 1 + 5 ) 1 2 ( 1 + 5 ) 1 2 ( 3 + 5 ) 1 2 ( 3 + 5 ) 1 1 1 2 ( 1 − 5 ) 1 2 ( 1 − 5 ) 1 2 ( 1 − 5 ) 1 2 ( 1 − 5 ) 1 2 ( 3 − 5 ) 1 2 ( 3 − 5 ) 1 − 1 1 2 ( 5 − 1 ) 1 2 ( 5 − 1 ) 1 2 ( 1 − 5 ) 1 2 ( 1 − 5 ) 1 2 ( 3 − 5 ) 1 2 ( 5 − 3 ) 1 − 1 1 2 ( − 1 − 5 ) 1 2 ( − 1 − 5 ) 1 2 ( 1 + 5 ) 1 2 ( 1 + 5 ) 1 2 ( 3 + 5 ) 1 2 ( − 3 − 5 ) 2 0 0 0 7 12 17 12 − 2 0 \begin{array}{|cccccccc|}
\hline
\mathbf{1} & \mathbf{2} & \mathbf{5} & \mathbf{6} & \mathbf{4} & \mathbf{3} & \mathbf{8} & \mathbf{7} \\
\hline
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) & \frac{1}{2} \left(1+\sqrt{5}\right) & \frac{1}{2} \left(3+\sqrt{5}\right) & \frac{1}{2} \left(3+\sqrt{5}\right) \\
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) & \frac{1}{2} \left(1-\sqrt{5}\right) & \frac{1}{2} \left(3-\sqrt{5}\right) & \frac{1}{2} \left(3-\sqrt{5}\right) \\
1 & -1 & \frac{1}{2} \left(\sqrt{5}-1\right) & \frac{1}{2} \left(\sqrt{5}-1\right) & \frac{1}{2} \left(1-\sqrt{5}\right) & \frac{1}{2} \left(1-\sqrt{5}\right) & \frac{1}{2} \left(3-\sqrt{5}\right) & \frac{1}{2} \left(\sqrt{5}-3\right) \\
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) & \frac{1}{2} \left(1+\sqrt{5}\right) & \frac{1}{2} \left(3+\sqrt{5}\right) & \frac{1}{2} \left(-3-\sqrt{5}\right) \\
2 & 0 & 0 & 0 & \frac{7}{12} & \frac{17}{12} & -2 & 0 \\
\hline
\end{array} 1 1 1 1 1 2 2 1 1 − 1 − 1 0 5 2 1 ( 1 + 5 ) 2 1 ( 1 − 5 ) 2 1 ( 5 − 1 ) 2 1 ( − 1 − 5 ) 0 6 2 1 ( 1 + 5 ) 2 1 ( 1 − 5 ) 2 1 ( 5 − 1 ) 2 1 ( − 1 − 5 ) 0 4 2 1 ( 1 + 5 ) 2 1 ( 1 − 5 ) 2 1 ( 1 − 5 ) 2 1 ( 1 + 5 ) 1 2 7 3 2 1 ( 1 + 5 ) 2 1 ( 1 − 5 ) 2 1 ( 1 − 5 ) 2 1 ( 1 + 5 ) 1 2 1 7 8 2 1 ( 3 + 5 ) 2 1 ( 3 − 5 ) 2 1 ( 3 − 5 ) 2 1 ( 3 + 5 ) − 2 7 2 1 ( 3 + 5 ) 2 1 ( 3 − 5 ) 2 1 ( 5 − 3 ) 2 1 ( − 3 − 5 ) 0 The numeric character table is the following
1 2 5 6 4 3 8 7 1.000 1.000 1.618 1.618 1.618 1.618 2.618 2.618 1.000 1.000 − 0.6180 − 0.6180 − 0.6180 − 0.6180 0.3820 0.3820 1.000 − 1.000 0.6180 0.6180 − 0.6180 − 0.6180 0.3820 − 0.3820 1.000 − 1.000 − 1.618 − 1.618 1.618 1.618 2.618 − 2.618 2.000 0 0 0 0.5833 1.417 − 2.000 0 \begin{array}{|rrrrrrrr|}
\hline
\mathbf{1} & \mathbf{2} & \mathbf{5} & \mathbf{6} & \mathbf{4} & \mathbf{3} & \mathbf{8} & \mathbf{7} \\
\hline
1.000 & 1.000 & 1.618 & 1.618 & 1.618 & 1.618 & 2.618 & 2.618 \\
1.000 & 1.000 & -0.6180 & -0.6180 & -0.6180 & -0.6180 & 0.3820 & 0.3820 \\
1.000 & -1.000 & 0.6180 & 0.6180 & -0.6180 & -0.6180 & 0.3820 & -0.3820 \\
1.000 & -1.000 & -1.618 & -1.618 & 1.618 & 1.618 & 2.618 & -2.618 \\
2.000 & 0 & 0 & 0 & 0.5833 & 1.417 & -2.000 & 0 \\
\hline
\end{array} 1 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 2 . 0 0 0 2 1 . 0 0 0 1 . 0 0 0 − 1 . 0 0 0 − 1 . 0 0 0 0 5 1 . 6 1 8 − 0 . 6 1 8 0 0 . 6 1 8 0 − 1 . 6 1 8 0 6 1 . 6 1 8 − 0 . 6 1 8 0 0 . 6 1 8 0 − 1 . 6 1 8 0 4 1 . 6 1 8 − 0 . 6 1 8 0 − 0 . 6 1 8 0 1 . 6 1 8 0 . 5 8 3 3 3 1 . 6 1 8 − 0 . 6 1 8 0 − 0 . 6 1 8 0 1 . 6 1 8 1 . 4 1 7 8 2 . 6 1 8 0 . 3 8 2 0 0 . 3 8 2 0 2 . 6 1 8 − 2 . 0 0 0 7 2 . 6 1 8 0 . 3 8 2 0 − 0 . 3 8 2 0 − 2 . 6 1 8 0 Representations of S L 2 ( Z ) SL_2(\mathbb{Z}) S L 2 ( Z )
This fusion ring does not provide any representations of S L 2 ( Z ) . SL_2(\mathbb{Z}). S L 2 ( Z ) .
Adjoint Subring
Elements 1 , 3 , 4 , 8 \mathbf{1}, \mathbf{3}, \mathbf{4}, \mathbf{8} 1 , 3 , 4 , 8 adjoint subring \(\text{Fib× \times × .
The upper central series is the following:
\(\text{FR}^{8,2}_{11} \underset{ \mathbf{1}, \mathbf{3}, \mathbf{4}, \mathbf{8} }{\supset} \text{Fib× \times ×
Universal grading
Each particle can be graded as follows: deg ( 1 ) = 1 ′ , deg ( 2 ) = 2 ′ , deg ( 3 ) = 1 ′ , deg ( 4 ) = 1 ′ , deg ( 5 ) = 2 ′ , deg ( 6 ) = 2 ′ , deg ( 7 ) = 2 ′ , deg ( 8 ) = 1 ′ \text{deg}(\mathbf{1}) = \mathbf{1}', \text{deg}(\mathbf{2}) = \mathbf{2}', \text{deg}(\mathbf{3}) = \mathbf{1}', \text{deg}(\mathbf{4}) = \mathbf{1}', \text{deg}(\mathbf{5}) = \mathbf{2}', \text{deg}(\mathbf{6}) = \mathbf{2}', \text{deg}(\mathbf{7}) = \mathbf{2}', \text{deg}(\mathbf{8}) = \mathbf{1}' deg ( 1 ) = 1 ′ , deg ( 2 ) = 2 ′ , deg ( 3 ) = 1 ′ , deg ( 4 ) = 1 ′ , deg ( 5 ) = 2 ′ , deg ( 6 ) = 2 ′ , deg ( 7 ) = 2 ′ , deg ( 8 ) = 1 ′ Z 2 \mathbb{Z}_2 Z 2
1 ′ 2 ′ 2 ′ 1 ′ \begin{array}{|ll|}
\hline
\mathbf{1}' & \mathbf{2}' \\
\mathbf{2}' & \mathbf{1}' \\
\hline
\end{array} 1 ′ 2 ′ 2 ′ 1 ′ Categorifications
Data
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