Csdrhrt

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 (or the book based on it) but for finite dimensional representations of $GL_n$. I feel like such a book or at least some expository notes should exist, but I have had zero luck finding any.

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 by Hersh and Lenart 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 this paper, might be helpful.