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Does anybody have book recommendations for reading A. Okounkov and A. Vershik's approach for complex irreducible representations of symmetric groups? Preferably, I am looking for a book that is intended for an undergraduate/first year graduate student who know basics of Representation theory.

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I am not aware of many books following the Okounkov-Vershik approach (which is a shame because it is way more enlightening, at least in some directions...). One option though is this book, Representation Theory of the Symmetric Groups: The Okounkov-Vershik Approach, Character Formulas, and Partition Algebras; you can tell by the title that it is exactly what you are looking for but I would warn you, it is not the easiest read...

The original paper is a bit dense to study but I guess that depends on your algebra background; these notes will help you if you decide to study from there a bit.

Another place to have a look, if you have access to it, is the book Graphs on Surfaces and their applications; read the appendix (by Zagier) on representation theory. Many libraries provide online access to the book, and Zagier manages to explain in a couple of pages the gist of what is going on in the Okounkov approach. At least you'll have a sort of roadmap for what's coming (the introduction of the original paper might seem a bit off-putting for all of us who lack Okounkov's and Vershik's insight).

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