I'm hoping to start an undergraduate on a project that involves understanding a bit of Gelfand-Tsetlin theory, and have been tearing my hair out looking for a good reference for them to look at. Basically what I would want is something at the level of Vershik-Okounkov – A new approach to the representation theory of symmetric groups. 2 (or the book Ceccherini-Silberstein, Scarabotti, and Tolli – Representation theory of the symmetric groups based on it) but for finite dimensional representations of $\mathrm{GL}_n$. I feel like such a book or at least some expository notes should exist, but I have had zero luck finding any.
[Math] n accessible exposition of Gelfand-Tsetlin theory
co.combinatoricslie-algebrasreference-requestrt.representation-theory
Best Answer
Good question. I've found this to be a difficult subject to get into myself. An abstract approach can seem arcane, but concrete constructions can be complicated and messy. You might try the paper Hersh and Lenart - Combinatorial construction of weight bases: The Gelfand–Tsetlin basis as a starting point. They take a concrete approach, which has the advantage that you can start computing with small examples relatively quickly. A disadvantage is that your student might miss the big picture of how all this fits into the general representation theory of classical Lie algebras. For that, perhaps the work of Molev, such as Molev - Gelfand–Tsetlin bases for classical Lie algebras, might be helpful.