Pseudo SU(2 ) 4 : FR 3 5 , 2 \text{Pseudo SU(2})_4:\ \text{FR}^{5,2}_{3} Pseudo SU(2 ) 4 : FR 3 5 , 2
Fusion Rules
1 2 3 4 5 2 1 4 3 5 3 4 2 + 5 1 + 5 3 + 4 4 3 1 + 5 2 + 5 3 + 4 5 5 3 + 4 3 + 4 1 + 2 + 5 \begin{array}{|lllll|}
\hline
\mathbf{1} & \mathbf{2} & \mathbf{3} & \mathbf{4} & \mathbf{5} \\
\mathbf{2} & \mathbf{1} & \mathbf{4} & \mathbf{3} & \mathbf{5} \\
\mathbf{3} & \mathbf{4} & \mathbf{2}+\mathbf{5} & \mathbf{1}+\mathbf{5} & \mathbf{3}+\mathbf{4} \\
\mathbf{4} & \mathbf{3} & \mathbf{1}+\mathbf{5} & \mathbf{2}+\mathbf{5} & \mathbf{3}+\mathbf{4} \\
\mathbf{5} & \mathbf{5} & \mathbf{3}+\mathbf{4} & \mathbf{3}+\mathbf{4} & \mathbf{1}+\mathbf{2}+\mathbf{5} \\
\hline
\end{array} 1 2 3 4 5 2 1 4 3 5 3 4 2 + 5 1 + 5 3 + 4 4 3 1 + 5 2 + 5 3 + 4 5 5 3 + 4 3 + 4 1 + 2 + 5
The fusion rules are invariant under the group generated by the following permutations:
{ ( 3 4 ) } \{(\mathbf{3} \ \mathbf{4})\} { ( 3 4 ) }
The following particles form non-trivial sub fusion rings
Particles
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 , 2 , 5 } \{\mathbf{1},\mathbf{2},\mathbf{5}\} { 1 , 2 , 5 }
Rep( D 3 ) : FR 2 3 , 0 \left.\text{Rep(}D_3\right):\ \text{FR}^{3,0}_{2} Rep( D 3 ) : FR 2 3 , 0
Quantum 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
1.73205 1.73205 1 . 7 3 2 0 5
3 \sqrt{3} 3
4 \mathbf{4} 4
1.73205 1.73205 1 . 7 3 2 0 5
3 \sqrt{3} 3
5 \mathbf{5} 5
2. 2. 2 .
2 2 2
D F P 2 \mathcal{D}_{FP}^2 D F P 2
12. 12. 1 2 .
12 12 1 2
Characters
The symbolic character table is the following
1 2 3 4 5 1 1 3 3 2 1 1 0 0 − 1 1 1 − 3 − 3 2 1 − 1 i − i 0 1 − 1 − i i 0 \begin{array}{|ccccc|}
\hline
\mathbf{1} & \mathbf{2} & \mathbf{3} & \mathbf{4} & \mathbf{5} \\
\hline
1 & 1 & \sqrt{3} & \sqrt{3} & 2 \\
1 & 1 & 0 & 0 & -1 \\
1 & 1 & -\sqrt{3} & -\sqrt{3} & 2 \\
1 & -1 & i & -i & 0 \\
1 & -1 & -i & i & 0 \\
\hline
\end{array} 1 1 1 1 1 1 2 1 1 1 − 1 − 1 3 3 0 − 3 i − i 4 3 0 − 3 − i i 5 2 − 1 2 0 0
The numeric character table is the following
1 2 3 4 5 1.000 1.000 1.732 1.732 2.000 1.000 1.000 0 0 − 1.000 1.000 1.000 − 1.732 − 1.732 2.000 1.000 − 1.000 1.000 i − 1.000 i 0 1.000 − 1.000 − 1.000 i 1.000 i 0 \begin{array}{|rrrrr|}
\hline
\mathbf{1} & \mathbf{2} & \mathbf{3} & \mathbf{4} & \mathbf{5} \\
\hline
1.000 & 1.000 & 1.732 & 1.732 & 2.000 \\
1.000 & 1.000 & 0 & 0 & -1.000 \\
1.000 & 1.000 & -1.732 & -1.732 & 2.000 \\
1.000 & -1.000 & 1.000 i & -1.000 i & 0 \\
1.000 & -1.000 & -1.000 i & 1.000 i & 0 \\
\hline
\end{array} 1 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 2 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 − 1 . 0 0 0 − 1 . 0 0 0 3 1 . 7 3 2 0 − 1 . 7 3 2 1 . 0 0 0 i − 1 . 0 0 0 i 4 1 . 7 3 2 0 − 1 . 7 3 2 − 1 . 0 0 0 i 1 . 0 0 0 i 5 2 . 0 0 0 − 1 . 0 0 0 2 . 0 0 0 0 0
Modular Data
This fusion ring does not have any matching S S S -and T T T -matrices.
Adjoint Subring
Particles 1 , 2 , 5 \mathbf{1}, \mathbf{2}, \mathbf{5} 1 , 2 , 5 , form the adjoint subring Rep( D 3 ) : FR 2 3 , 0 \left.\text{Rep(}D_3\right):\ \text{FR}^{3,0}_{2} Rep( D 3 ) : FR 2 3 , 0 .
The upper central series is the following:
Pseudo SU(2 ) 4 ⊃ 1 , 2 , 5 Rep( D 3 ) \text{Pseudo SU(2})_4 \underset{ \mathbf{1}, \mathbf{2}, \mathbf{5} }{\supset} \left.\text{Rep(}D_3\right) Pseudo SU(2 ) 4 1 , 2 , 5 ⊃ Rep( D 3 )
Universal grading
Each particle can be graded as follows: deg ( 1 ) = 1 ′ , deg ( 2 ) = 1 ′ , deg ( 3 ) = 2 ′ , deg ( 4 ) = 2 ′ , deg ( 5 ) = 1 ′ \text{deg}(\mathbf{1}) = \mathbf{1}', \text{deg}(\mathbf{2}) = \mathbf{1}', \text{deg}(\mathbf{3}) = \mathbf{2}', \text{deg}(\mathbf{4}) = \mathbf{2}', \text{deg}(\mathbf{5}) = \mathbf{1}' deg ( 1 ) = 1 ′ , deg ( 2 ) = 1 ′ , deg ( 3 ) = 2 ′ , deg ( 4 ) = 2 ′ , deg ( 5 ) = 1 ′ , where the degrees form the group Z 2 \mathbb{Z}_2 Z 2 with multiplication table:
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
Download links for numeric data: