In physics, the wave function is a mathematical function $\psi: \mathbb{R}^3 \to \mathbb{C}$. In the discussion of fermions and bosons we can talk about how the wave function behave under the interchange of two particles. There are two fundamental cases:
$$ \psi(x,y) = \pm \psi(y,x)$$
If there is a "+" we get bosons, in the case of "-" we get a fermion. Indeed we can construct functions in 3 variables which do the same thing:
$$ \psi(x,y,z) = \psi(y,z,x) = \psi(z,y,x) = - \psi(y,x,z) = - \psi(x,z,y)= - \psi(z,y,x)$$
Indeed, any function in two variables can be split into the symmetric and anti-symmetric part:
$$ \psi(x,y) = \frac{1}{2}\Big[ \underbrace{\psi(x,y) + \psi(y,x)}_S \Big] + \frac{1}{2}\Big[ \underbrace{\psi(x,y) - \psi(y,x)}_{A} \Big]$$
In terms of representation theory we are showing a very special case of Schur-Weyl duality (or plethysm? I forget the name): $V^2 = \wedge^2 \,V \oplus \mathrm{Sym}^2(V) $
We can write two different projection operators. One is "take the symmetric part":
\begin{eqnarray*}
S\psi(x,y) &=& \tfrac{1}{2} \big[ \psi(x,y) + \psi(y,x) \big] \\
A\psi(x,y) &=& \tfrac{1}{2} \big[ \psi(x,y) - \psi(y,x) \big]
\end{eqnarray*}
These two projection operators are examples of Young symmetrizers. For 3 or more particles there are more examples, using young tableaux.
I am stopping here to save my work in case my computer crashes (as it sometimes does).
