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