The key to identical particles is understanding the configuration space properly. Basically, instead of ordered n-tuples (R^3N), one can use sets of size n whose elements are points in R^3. This immediately eliminates everything except the "symmetric" wave functions. To get the anti-symmetric functions needed for fermions, one needs to extend the theories under consideration a bit. One could use a covering space approach, demanding a suitable projection of the trajectories downwards or one can approach it from the perspective of having a twisted bundle for the value space of the wave function, one in which it looks locally like complex-valued wave functions, but globally it has a parallel transport action that leads to sign change around a closed path that suitably permutes the set.

The nice thing about the parallel transport point of view is the extension to spin. Again, it will look locally like the usual tensor product, but by using an index set for the tensor product being just the set of positions itself, one can again get a non-trivial parallel transport that does just what one would expect.

This is what restricts the permissible wave functions.

One reference for this is my own thesis from about 20 years ago:

http://jostylr.com/thesis.pdf

There were some papers that were published from that, but they mainly focused on the general topological story rather than having very much in the way of identical particles.

One thing we never did accomplish from this point of view was explaining why fermions went with half spins and bosons integer spins. But at least we did get it reduced to just the two choices.