Background: The arena is fixed particle number nonrelativistic quantum mechanics. The state space is $$ \mathbf{H}(1)=\mathcal H\otimes\mathcal S, $$ where $\mathcal H$ is an "orbital" state space ($L^2(\mathbb R^3)$ if one'd like), and $\mathcal S$ is the spin space, which for a particle of spin $s$ is identifiable with $\mathbb C^{2s+1}$.

Because in this case it is (in my opinion) more tractable notationally, I will identify $\mathcal H$ with $L^2(\mathbb R^3)$ (position-basis wavefunctions), and a basis for $\mathcal S$ is $\left|\sigma\right\rangle$, where $\sigma=-s,-s+1/2,...,+s$. We can identify $\left|\sigma\right\rangle$ with the $\sigma$th standard basis vector of $\mathbb C^{2s+1}$.

A general 1-particle state is then $$ \Psi(\mathbf x)=\sum_\sigma\psi_\sigma(\mathbf x)\left|\sigma\right\rangle, $$ which is just a $2s+1$-component wavefunction.

The $k$-particle fermionic state space is $$ \mathbf H_A(k)=\bigwedge^k(\mathcal H\otimes\mathcal S), $$ and the $k$-particle bosonic state space is $$ \mathbf H_S(k)=\bigvee^k(\mathcal H\otimes\mathcal S). $$

In case $k=2$, the construction of a general state is easy. The "unsymmetric" state space is $\mathcal H\otimes\mathcal S\otimes\mathcal H\otimes\mathcal S\simeq \mathcal H\otimes\mathcal H\otimes\mathcal S\otimes\mathcal S$, and in the product spin space $\mathcal S^2\equiv\mathcal S\otimes\mathcal S$, and we may easily construct a basis, it is $$\Sigma_{\sigma_1\sigma_2}\equiv \left|\sigma_1\sigma_2\right\rangle\equiv\left|\sigma_1\right\rangle\otimes \left|\sigma_2\right\rangle, $$ which we may identify with $(2s+1)\times(2s+1)$ square matrices, particularily, $\Sigma_{\sigma_1\sigma_2}$ is the matrix whose $(\sigma_1,\sigma_2)$th element is 1, and the rest is zero.

A general "unsymmetric" state is $$ \Psi(\mathbf x_1,\mathbf x_2)=\sum_{\sigma_1,\sigma_2}\psi_{\sigma_1\sigma_2}(\mathbf x_1,\mathbf x_2)\Sigma_{\sigma_1\sigma_2}. $$

To find symmetric or skew-symmetric states, one may decompose $$ \Sigma_{\sigma_1\sigma_2}=S_{\sigma_1\sigma_2}+A_{\sigma_1\sigma_2}, $$ where $S_{\sigma_1\sigma_2}=\Sigma_{(\sigma_1\sigma_2)}$ and $A_{\sigma_1\sigma_2}=\Sigma_{[\sigma_1,\sigma_2]}$. Here I am ignoring the normalizations of the bases, as that is conceptually not relevant. There are $2s^2+3s+1$ independent $S_{\sigma_1\sigma_2}$ states, and $2s^2+s$ independent $A_{\sigma_1\sigma_2}$ states and together they span $\mathcal S^2$, so we can express a general symmetric state as $$ \Psi_S(\mathbf x_1,\mathbf x_2)=\sum_{\sigma_1\le\sigma_2}\psi^S_{\sigma_1\sigma_2}(\mathbf x_1,\mathbf x_2)S_{\sigma_1\sigma_2}+\sum_{\sigma_1<\sigma_2}\psi^A_{\sigma_1\sigma_2}(\mathbf x_1,\mathbf x_2)A_{\sigma_1,\sigma_2}, $$ where the $\psi^S_{\sigma_1\sigma_2}(\mathbf x_1,\mathbf x_2)$ wavefunctions are symmetric with respect to the exchange of the positions $\mathbf x_1,\mathbf x_2$ and the wavefunctions $\psi^A_{\sigma_1\sigma_2}(\mathbf x_1,\mathbf x_2)$ are skew.

A general skew-symmetric state is expressed then by the same combination but with the $\psi^S_{\sigma_1\sigma_2}(\mathbf x_1,\mathbf x_2)$ being skew in the position vectors and the functions $\psi^A_{\sigma_1\sigma_2}(\mathbf x_1,\mathbf x_2)$ being symmetric in the position vectors.


Assume I want to do the same thing but with $k$ particles instead of 2. The situations seems much more complicated, mainly because the spin bases $$ \Sigma_{\sigma_1...\sigma_k} $$ now cannot be split into totally symmetric and skew-symmetric parts while still spanning the set of all "unsymmetric" spin states.

So, if I want to construct all bosonic and all fermionic $k$-particle states, what is the analogue of what I have done with 2-particle states?

I have a feeling this might actually be quite untractable generally (we want to have the decomposition of $\bigwedge V\otimes W$ into direct sums of exterior (or symmetric) products, which by this answer here: https://math.stackexchange.com/questions/211561/exterior-power-of-a-tensor-product/553405 seems untractable), however I assume this is a very important situation for applied quantum mechanics, so in this case, how to construct symmetry-respecting states?


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