Intuition behind the construction of Young Symmetrizer

I've been studying the representation theory of groups from Tung's "Group Theory in Physics." I understand Young symmetrizers of different Young diagrams are essentially primitive idempotents in the group algebra of the symmetric group and then all inequivalent minimal left ideals as well as all inequivalent irreducible representation can be obtained.

However, the construction seems unintelligible to me, while the property of Young symmetrizers is so striking. What is the idea behind the construction?

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).