How are spin network edges related to anti-symmetric projectors on the Hilbert space of the fundamental rep of SU(2)?

Question

In the paper here https://arxiv.org/pdf/gr-qc/9905020.pdf we see an introduction to Spin-networks of the original Penrose type i.e an undirected open graph whose edges have labels that are irreducible representations of SU(2).

In particular given a set of label 1 edges (that represent spin 1/2), here shown with no label, it is said that one can create higher label wires as follows:

Where $S_n$ is the space of permutations of n and $|\sigma|$ is the number of 'wire crossings' induced by said permutation. Looking at this it appears to be a projector from the Hilbert space $\otimes_n H^{ 2}$ to its anti-symmetric subspace - essentially a direct generalisation of how this

sends $H^{2}\otimes H^{2}$ to its anti-symmetric subspace.

My question is how does this gel with representation theory? Is it the case that any irreducible representation can somehow be seen in terms of an anti-symmetric projector on a tensor product space of Hilbert spaces acted on by an appropriate number of copies the fundamental rep of SU(2)?

Doubtless my relative weakness in representation theory is apparent. I'm approaching this problem due to some overlap with quantum computing so the more things are tied to concrete things like qubit basis', and fundamental SU(2) operations the more likely I will understand what is going on.

Answer 1

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On page 4, the author of https://arxiv.org/pdf/gr-qc/9905020.pdf states that their swap $$ ⛌_{AC}^{BD}: A \otimes C \rightarrow B \otimes D $$ is not the usual swap $\delta_A^D\delta_C^B$ which we all know and love, but rather $-1$ times the usual swap, i.e. $$ ⛌_{AC}^{BD} := -\delta_A^D\delta_C^B $$ This is to cure problems with topological deformation of string diagrams introduced by their particular choice of caps as $\epsilon_{AB}$ and cups as $\epsilon^{AB}$, as described on page 3: $$ \epsilon_{AB} = \epsilon^{AB} = \left( \begin{matrix} 0 & 1 \\ -1 & 0 \end{matrix} \right) $$

In particular, this means that the map $\left(\mathbb{C}^2\right)^{\otimes n} \rightarrow \left(\mathbb{C}^2\right)^{\otimes n}$ described below: $$ \frac{1}{n!}\sum_{\sigma \in S_n} \operatorname{sgn}(\sigma) U_\sigma $$ where $U_\sigma$ implements $\sigma$ using the swap $⛌_{AC}^{BD}$: $$ U_\sigma(e_1 \otimes ... \otimes e_n) = \operatorname{sgn}(\sigma) e_{\sigma(1)} \otimes ... \otimes e_{\sigma(n)} $$ is actually the projector on the symmetric subspace $\operatorname{S}^n(\mathbb{C}^2)$, not on the antisymmetric one: $$ \left(\frac{1}{n!}\sum_{\sigma \in S_n} \operatorname{sgn}(\sigma) U_\sigma\right) (e_1 \otimes ... \otimes e_n)\\ = \frac{1}{n!}\sum_{\sigma \in S_n} \operatorname{sgn}(\sigma)^2 e_{\sigma(1)} \otimes... \otimes e_{\sigma(n)}\\ = \frac{1}{n!}\sum_{\sigma \in S_n} e_{\sigma(1)} \otimes... \otimes e_{\sigma(n)} $$ That gives the usual symmetric product formulation of the irreps of SU(2).

Answer 2

Stefano gave a very nice answer, but let me add a couple things. Penrose's description is indeed more topological than representation theoretic. Loops which normally are traces of the identity should evaluate to $+2$, but he assigns to them the value $-2$. Penrose's lines do not really carry indices: spin networks with free legs cannot be directly understood as some tensors with indices carried by the free legs. Finally, Penrose uses antisymmetrizers where one would expect symmetrizers. Essentially, Penrose's description is spin networks for $SU(-2)$. There is precise way to relate that to the more standard $SU(2)$ multilinear/tensor algebra and representation theory. See the negative dimensionality theorem in Section 4 of my JKTR article "On the volume conjecture for classical spin networks". The name comes from the book "Group Theory" by Predrag Cvitanović. Note that Penrose's point of view has some advantages like simplifying some computations. See the derivation of Racah's single sum formula for the $6j$ symbol in the book by Cvitanović.