SU(2 ) 4 : FR 3 5 , 0 \text{SU(2})_4:\ \text{FR}^{5,0}_{3} SU(2 ) 4 : FR 3 5 , 0
Fusion Rules
1 2 3 4 5 2 1 4 3 5 3 4 1 + 5 2 + 5 3 + 4 4 3 2 + 5 1 + 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{1}+\mathbf{5} & \mathbf{2}+\mathbf{5} & \mathbf{3}+\mathbf{4} \\
\mathbf{4} & \mathbf{3} & \mathbf{2}+\mathbf{5} & \mathbf{1}+\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 1 + 5 2 + 5 3 + 4 4 3 2 + 5 1 + 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 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 , 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
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
1.73205 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 1 − 1 0 1 − 1 − 1 1 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 & 1 & -1 & 0 \\
1 & -1 & -1 & 1 & 0 \\
\hline
\end{array} 1 1 1 1 1 1 2 1 1 1 − 1 − 1 3 3 0 − 3 1 − 1 4 3 0 − 3 − 1 1 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 − 1.000 0 1.000 − 1.000 − 1.000 1.000 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 & -1.000 & 0 \\
1.000 & -1.000 & -1.000 & 1.000 & 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 − 1 . 0 0 0 4 1 . 7 3 2 0 − 1 . 7 3 2 − 1 . 0 0 0 1 . 0 0 0 5 2 . 0 0 0 − 1 . 0 0 0 2 . 0 0 0 0 0
Representations of S L 2 ( Z ) SL_2(\mathbb{Z}) S L 2 ( Z )
The matching S S S -matrices and twist factors are the following
S S S -matrix
Twist factors
1 2 3 ( 1 1 3 3 2 1 1 − 3 − 3 2 3 − 3 3 − 3 0 3 − 3 − 3 3 0 2 2 0 0 − 2 ) \frac{1}{2 \sqrt{3}}\left(\begin{array}{ccccc} 1 & 1 & \sqrt{3} & \sqrt{3} & 2 \\ 1 & 1 & -\sqrt{3} & -\sqrt{3} & 2 \\ \sqrt{3} & -\sqrt{3} & \sqrt{3} & -\sqrt{3} & 0 \\ \sqrt{3} & -\sqrt{3} & -\sqrt{3} & \sqrt{3} & 0 \\ 2 & 2 & 0 & 0 & -2 \\\end{array}\right) 2 3 1 ⎝ ⎜ ⎜ ⎜ ⎜ ⎛ 1 1 3 3 2 1 1 − 3 − 3 2 3 − 3 3 − 3 0 3 − 3 − 3 3 0 2 2 0 0 − 2 ⎠ ⎟ ⎟ ⎟ ⎟ ⎞
( 0 , 0 , 3 8 , − 1 8 , − 1 3 ) ( 0 , 0 , − 1 8 , 3 8 , − 1 3 ) ( 0 , 0 , 1 8 , − 3 8 , 1 3 ) ( 0 , 0 , − 3 8 , 1 8 , 1 3 ) \begin{array}{l}\left(0,0,\frac{3}{8},-\frac{1}{8},-\frac{1}{3}\right) \\\left(0,0,-\frac{1}{8},\frac{3}{8},-\frac{1}{3}\right) \\\left(0,0,\frac{1}{8},-\frac{3}{8},\frac{1}{3}\right) \\\left(0,0,-\frac{3}{8},\frac{1}{8},\frac{1}{3}\right)\end{array} ( 0 , 0 , 8 3 , − 8 1 , − 3 1 ) ( 0 , 0 , − 8 1 , 8 3 , − 3 1 ) ( 0 , 0 , 8 1 , − 8 3 , 3 1 ) ( 0 , 0 , − 8 3 , 8 1 , 3 1 )
1 2 3 ( 1 1 3 3 2 1 1 − 3 − 3 2 3 − 3 − 3 3 0 3 − 3 3 − 3 0 2 2 0 0 − 2 ) \frac{1}{2 \sqrt{3}}\left(\begin{array}{ccccc} 1 & 1 & \sqrt{3} & \sqrt{3} & 2 \\ 1 & 1 & -\sqrt{3} & -\sqrt{3} & 2 \\ \sqrt{3} & -\sqrt{3} & -\sqrt{3} & \sqrt{3} & 0 \\ \sqrt{3} & -\sqrt{3} & \sqrt{3} & -\sqrt{3} & 0 \\ 2 & 2 & 0 & 0 & -2 \\\end{array}\right) 2 3 1 ⎝ ⎜ ⎜ ⎜ ⎜ ⎛ 1 1 3 3 2 1 1 − 3 − 3 2 3 − 3 − 3 3 0 3 − 3 3 − 3 0 2 2 0 0 − 2 ⎠ ⎟ ⎟ ⎟ ⎟ ⎞
( 0 , 0 , − 1 8 , 3 8 , 1 3 ) ( 0 , 0 , 3 8 , − 1 8 , 1 3 ) ( 0 , 0 , − 3 8 , 1 8 , − 1 3 ) ( 0 , 0 , 1 8 , − 3 8 , − 1 3 ) \begin{array}{l}\left(0,0,-\frac{1}{8},\frac{3}{8},\frac{1}{3}\right) \\\left(0,0,\frac{3}{8},-\frac{1}{8},\frac{1}{3}\right) \\\left(0,0,-\frac{3}{8},\frac{1}{8},-\frac{1}{3}\right) \\\left(0,0,\frac{1}{8},-\frac{3}{8},-\frac{1}{3}\right)\end{array} ( 0 , 0 , − 8 1 , 8 3 , 3 1 ) ( 0 , 0 , 8 3 , − 8 1 , 3 1 ) ( 0 , 0 , − 8 3 , 8 1 , − 3 1 ) ( 0 , 0 , 8 1 , − 8 3 , − 3 1 )
Adjoint Subring
Elements 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:
SU(2 ) 4 ⊃ 1 , 2 , 5 Rep( D 3 ) \text{SU(2})_4 \underset{ \mathbf{1}, \mathbf{2}, \mathbf{5} }{\supset} \left.\text{Rep(}D_3\right) 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
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