I haven't found anywhere that discusses multi-particle quantum mechanics exactly like this, and I'm not sure the correct thing to Google. If anyone has any pointers to a reference that might help, I'd appreciate it!
In first-quantization, the Hamiltonian operator $\hat{H}$ acts on wavefunctions of $N$ variables $\psi(x_1,...,x_N)$. For example, a standard $N$-body Hamiltonian might look like $\hat{H}=-\sum_i\frac{1}{2m}\frac{\partial^2}{\partial x_i^2}+\frac{1}{2}\sum_{i\neq j}V(x_i-x_j)$. If this is a Hamiltonian describing identical particles, it must commute with the permutation operators defined by $\hat{P}_\sigma\psi(x_1,...,x_N)\equiv\psi(x_{\sigma(1)},...,x_{\sigma(N)})$.
If we consider the set of all $\hat{H}$ that commute with the permutation operators, these $\hat{H}$ form a semigroup. It's obvious that this semigroup has at least two invariant subspaces. The antisymmetric subspace (wavefunctions for Fermions) is preserved, and the symmetric subspace (wavefunctions for Bosons) is also preserved under this semigroup.
My question is: How many invariant subspaces are there for general $N$? For $N=2$, the symmetric and antisymmetric subspaces span the space of all functions, so there are only two invariant subspaces. For $N>2$, there's at least a third subspace, the set of functions orthogonal to all antisymmetric and symmetric functions, that is clearly preserved by the semigroup. But it's not clear to me if this space breaks into smaller invariant subspaces, or if the semigroup mixes all the states orthogonal to the symmetric and antisymmetric subspaces.
I've asked a similar question on the math stackexchange, where I've phrased it in terms of finite-dimensional representation theory, if anyone finds that simpler to think about than symmetries of wavefunctions.
