This came up during a Particle Physics discussion :

Here is a proof that the wavefunction of n identical particles has to be either completely symmetric or anti-symmetric.

Wavefunction ψ=ψ(a1,a2,……..,an)

Let Pij be the operator which flips particles i and j

Now indistinguishability requires

Pijψ=c ψ

Since flipping twice results in the same wavefunction

Pij Pij ψ = ψ

Implies c=1 or -1

Now consider this relation

Pij=Pik Pkj Pki

One can easily veryify that this true by looking at the action of Pij on the ordrerd set (i,j,k).

Since Pik Pik=1


Extending this simlarly one gets

Pij = Pik = Pkl for any k,l

Thus Pfg = c for all f,g

Thus the wavefunction has to be completely symmetric or antisymmetric.


PS: I took the liberty of tidying up the notation, hope I didn't make any mistake/ change your argument. –Shanth