This is a rather soft question. I was wondering if someone could explain on a fundamental and intuitive level, what the Okounkov-Vershik approach to representation theory of $S\_n$ is all about. It's one thing to read a few chapters of a book or a paper, but it's another to try and understand the wider picture of the theory.


For reference, here is one of the main papers:
<http://www.mat.univie.ac.at/~esiprpr/esi333.pdf>


Any explanations would be gratefully received.



Okounkov-Vershik approach to representation theory of $S_n$
First, I'll note that they provide a pretty clear explanation of their motivation on page two of that paper. So it would strengthen your question if you indicated you had read that, and what about it you found unsatisfying. 


My own feeling is that the key observation of their approach is that the space of elements of $\mathbb{Q}[S\_n]$ that commute with $\mathbb{Q}[S\_{n-1}]$ is generated (**edit**: over the center of $\mathbb{Q}[S\_{n-1}]$) by a single element, the Jucys-Murphy element $X\_n=\sum\_{i<n} (i,n)$. This adds an additional structure of the functors of induction and restriction, which is the endomorphism of multiplication by $X\_n$: on a restriction, you act in the obvious way, and on an induction, you send $g\otimes v\mapsto gX\_n\otimes v$. 


Thus, you can decompose these functors according to the eigenvalues of $X\_n$, which gives you much more control over the situation. Remarkably, these subfunctors corresponding to different eigenvalues actually send simple modules to simple modules, and you can naturally pull the description of simples in terms of Young diagrams out of their interactions.


