I have been reading about non-abelian fluxons in John Preskill's lectures notes on topological quantum computing and I do not understand how he deduced the fusion rules for fluxons in the example he gave.

$S_3$ fluxons example

Preskill's example is non-abelian fluxons of the group $G = S_3$, the permutation group on three elements. As shown on page 31, the total list of particles in the theory is given by

\begin{array}{|c|c|c|c|} \hline \text{Type}& \text{Flux} & \text{Charge} & \text{Dim} \\ \hline A & e& [+]& 1\\ \hline B & e & [-] &1\\ \hline C & e & [2]&2\\ \hline D & (12) & [+] &3\\ \hline E & (12) &[-] &3\\ \hline F & (123) &[1] &2\\ \hline G & (123) &[\omega] &2\\ \hline H & (123) & [\omega] &2\\ \hline \end{array}

where the flux labels are the possible conjugacy classes $\alpha$ of $S_3$ and the charge labels are the irreps of normaliser $N(\alpha)$ of the corresponding conjugacy class. It is stated that when we fuse two particles together we apply the following rule:

The flux of the composite can belong to any of the conjugacy classes that can be obtained as a product of representatives of the classes that label the two constituents. Finding the charge of the composite is especially tricky, as we must decompose a tensor product of representations of two different normalizer groups as a sum of representations of the normalizer of the product flux.

He provides the example that if we fuse two $D$ particles, we have in equation (9.47)

$$ D \times D = A + C + F + G + H $$

My question

I do not undestand how this fusion rule was deduced. Preskill states that the possible charges of the composite must be formed by taking tensor products of the charges of the constituent particles. For the case of $D$ particles, the charge of these particles is the trivial representation $[+]$. If we take a tensor product of two trivial representations, it should surely form another trivial representation that will not decompose into a direct sum of irreps, i.e. $[ + ] \otimes [+] = [+]$, so how do we get the sum above?

Preskill does not give much detail on exactly what the kinematics of these particles are. I would expect for a composite fluxon with flux labelled by a conjugacy class $\alpha$ and charge given by a representation of the corresponding normaliser $R_{(\alpha)}^i$, the Hilbert space that describes this system will be something like

$$ H_{\alpha}^i = \mathcal{H}_\alpha \otimes R_{(\alpha)}^i $$

where $\mathcal{H}_\alpha$ describes the flux degrees of freedom and $R_{(\alpha)}^i$ describes the charge degrees of freedom. So for example for a particle of type $D$ I would have

$$ H_D = \mathcal{H}_{(12)} \otimes R_{(12)}^{[+]}$$

so for fusion of two $D$ fluxons, I would write

$$ H_D \otimes H_D = (\mathcal{H}_{(12)} \otimes R_{(12)}^{[+]}) \otimes (\mathcal{H}_{(12)} \otimes R_{(12)}^{[+]}) \stackrel{?}{=} H_A \oplus H_C \oplus H_F \oplus H_G \oplus H_H$$

which I expect will decompose into irreps much like (9.47) above? Is this what John Preskill did to deduce the fusion rule?


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