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It's well known that Young symmetrizer is a fundamental tool in the representation theory of symmetric groups.

For instance, for every Young diagram $\lambda\vdash n$, we construct a Young tableau $T_\lambda$ by filling in the numbers $1,2,\ldots,n$ in increasing order, left to right, top to bottom. Then we define the two Young projectors

$$ a_{\lambda}:=\sum_{g\in P_{T_\lambda}}g\;\;,\quad b_{\lambda}:=\sum_{g\in Q_{T_\lambda}}(-1)^{g}g $$

where $P_{T_\lambda}$ is the row stabilizer subgroup and $Q_{T_\lambda}$ is the column stabilizer subgroup. The Young symmetrizer is then defined as $c_\lambda:=a_\lambda b_\lambda$ and the Specht module is $V_\lambda:=\mathbb{C}[S_n]c_\lambda$. Every irreducible complex representations of $S_n$ is isomorphic to $V_\lambda$ for a unique $\lambda$.

However, I don’t quite understand how the definition of the Young symmetrizer is derived. Is there a more natural way to introduce this element?

To avoid the somewhat unnatural Young symmetrizer, I came across a beautiful and elementary approach to the theory of representation of $S_n$, developed by Okounkov and Vershik. In this framework, every irreducible representation $V^\lambda$ of $S_n$ is also associated with a Young diagram $\lambda$. However, this raises additional questions: Is $V^\lambda$ in Okounkov-Vershik theory isomorphic to the Specht module $V_\lambda = \mathbb{C}[S_n] c_\lambda$? Can we rediscover the Young symmetrizer $c_\lambda$ in Okounkov-Vershik theory?

The sole answer to a similar question on MSE seems to treat Young symmetrizer as the key point to Schur-Weyl duality. Another question explores the characterization of the Specht module, which may provide additional insights.

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    $\begingroup$ There are many ways to define Specht modules (see the various "avatars" in my notes, and there are more I plan to add). The Okounkov-Vershik one is one of the less natural, as it only works in characteristic $0$ to the best of my knowledge. But yes, it is isomorphic to the Specht module. This should be in the book that I cite as [CSScTo10] in the above-linked notes. $\endgroup$
    – darij grinberg
    Commented 2 days ago
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    $\begingroup$ [CSScTo10] on this matter follows Vershik's "A new approach to the representation theory of the symmetric groups, III." However, there is a gap in Vershik's argument, repeated in [CSScTo10]. The gap is explained in the bachelor's thesis of Matthew J Hall at the University of Arizona. $\endgroup$
    – Joshua Mundinger
    Commented 2 days ago
  • $\begingroup$ Isn't the idea that if you use just the row stabilizer, you get the action on words, corresponding to $h_\lambda$, which contains $s_\mu$ only if $\mu \leq \lambda$ in dominance order, whereas if you use just the column stabilizer you get the related $e_{\lambda'}$, which contains $s_{\mu'}$ only if $\mu' \leq \lambda'$? Then if you do both since conjugation reverses dominance order the only thing left is $\mu = \lambda$? I confess I never filled in the details. $\endgroup$
    – Joshua P. Swanson
    Commented yesterday
  • $\begingroup$ If I'm not mistaken, the symmetrisers $c_\lambda = a_\lambda \, b_\lambda$ have the drawback that they are not orthogonal. (I don't remember when the first counter-example shows up; it might be $n=6$ where the dominance order is no longer a total ordering.) The construction used by Okounkov--Vershik (which I believe was essentially already found by Young, On quantitative substitutional analysis VI, 1931; see p208 and 214 therein) gives orthogonal projections and even matrix units inside $\mathbb{C}S_n$ , cf mathoverflow.net/q/441536. $\endgroup$
    – Jules Lamers
    Commented yesterday

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