# 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?

• Assuming you're still around to read this comment, you could ask this over at MathOverflow if my bump and bounty offer don't get it any new eyes in the next week. – anon Aug 22 '15 at 6:06
• @anon Thanks a lot for your attention. :) – Eli4ph Aug 23 '15 at 4:47

## 1 Answer

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).

• hep.caltech.edu/~fcp/math/groupTheory/young.pdf – cactus314 Aug 23 '15 at 16:59
• Yes, the cases of symmetrizing and antisymmetrizing are comparatively obvious and intuitive - they can be discovered just playing around with algebra. Can you tell a plausible story for how we might discover general Young symmetrizers though, or give a historical account? That is the question here, after all. (Just read your last comment. I guess I'll reserve judgment until it's finished.) – anon Aug 24 '15 at 2:03
• @anon sorry I have been occupied. – cactus314 Aug 24 '15 at 2:04