TITLE: Some interesting and elementary topics with connections to the representation theory?
QUESTION [17 upvotes]: I'm going to give a talk to talented high school seniors (for nearly 1.25-1.75 hours, maybe a little bit longer). They know some  abstract algebra (groups, rings, modules...), linear algebra (including the Jordan normal form, nilpotent operators, etc.), a real (and basics of complex) calculus, some elementary topology (basics from the general topology, fundamental groups and maybe something about manifolds or vector fields), basics of algebraic geometry and a lot of combinatorial stuff (including the graph theory and generating functions).
My main interest is the representation theory, so I'd like to discuss an algebraic topic connected with this branch of mathematics. It'll be cool if this topic contains some beautiful combinatorial constructions. Despite this, some level of abstraction is required... I think that it mustn't also be too famous (so the standard things like representations of symmetric groups (even in Vershik-Okounkov approach) or the basics of Lie theory aren't acceptable). Ideally, it has to be new to me...
Can someone give me a piece of advice about which topic can be chosen in this situation?
I thought about the cluster algebras (maybe in the flavor of the first pages of the paper [1]?)... But are there any elementary applications of them? (And, by the way, are there any classical monographs about this subject?)
[1] -- http://ovsienko.perso.math.cnrs.fr/Publis/FriezeNew1.pdf
UPD: There is a question which looks like similar to this on MO. Namely, Fun applications of representations of finite groups . But, of course, it's very different. The reason is that I'm more interested in representations of more complicated than finite groups structures like, for example, quivers or Lie algebras. So the other question isn't relevant: its topic is too narrow.

REPLY [3 votes]: One example of an elementary application of cluster algebras is the proof that the Somos-4 and Somos-5 sequences, which are defined by a simple recursion, are integral. This is so because the entries of these sequences are cluster variables with the initial cluster variables specialized to 1, so it follows from the Laurent phenomenon. See Fomin and Zelevinsky's paper The Laurent Phenomenon.
One wouldn't have time to prove the Laurent phenomenon, but one can demonstrate it, and it is quite striking.