I'm hoping to start an undergraduate on a project that involves understanding a bit of GelfandTsetlin 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 VershikOkounkov  A new approach to the representation theory of symmetric groups. 2 (or the book CeccheriniSilberstein, 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.

3Perhaps such a reference doesn't exist, especially if you and Tim have both looked for it and not found it. If it did exist, it would probably be within the scope of the Graduate Journal of Mathematics, gradmath.org, which "publishes original work as well as expository work [that] helps make more widely accessible significant mathematical ideas, constructions or theorems." One option would be to have your student write up the sort of thing you're looking for and submit it to GJM. The website says "High quality senior theses will find GJM to be a great venue" – David White Nov 14 at 18:00

How about the Allen's notes from when he taught Lie groups in 20012002? I remember there was one on GelfandTsetlin (or GelfandCetlin as Allen spelled it). Those notes aren't online anymore, but hopefully Allen still has a copy. – Joel Kamnitzer Dec 8 at 4:40

@JoelKamnitzer I found some notes on Allen’s website, but they were not really the style I was looking for. In particular, they assumed a lot of familiarity with the WCF in a way that didn’t really match what I had in mind. – Ben Webster♦ Dec 11 at 1:42
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.