I have just read a big part of John Baez’s nice article Spin network states in Gauge theory. The definitions are quite clear in that article. However, there is a part which is not explained explicitly there, in my humble opinion. Let us say you have a spin network corresponding to a compact connected Lie group $G$ (the gauge group), which is a finite oriented graph with each edge labeling an irreducible representation of $G$ and each vertex $v$ labeling an intertwining operator (for the action of $G$) from the tensor product of the (irreducible) representations labeled by the set of all incoming edges at $v$ to the tensor product of the reps labeled by the set of all outgoing edges at $v$.

What I would like to understand though, is how to regard a spin network as an element of $mathrm{L}^2(mathcal{A}/mathcal{G})$, where $mathcal{A}$ denotes the space of connections on the graph (here regarded as parallel transport maps associated to each edge, please see the article for more detail) and $mathcal{G}$ is the group of gauge transformations which acts on $mathcal{A}$.

Given a spin network and a “connection” $A$ on the graph, how do we “evaluate” the spin network on the connection $A$, or rather on the equivalence class of $A$ under the group $mathcal{G}$ of gauge transformations? The description in that article is via identifications and the Peter-Weyl theorem. Could someone perhaps spell it out for me in more concrete terms?