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.
Best Answer
Here three suggestions:
-An applications of representation theory to chemistry: "The Representation-Theory for Buckminsterfullerene" by Gordon James: https://www.sciencedirect.com/science/article/pii/S0021869384712130?via%3Dihub
-For a mix with probability theory the book Harmonic Analysis on Finite Groups: Representation Theory, Gelfand Pairs and Markov Chains by Tullio Ceccherini-Silberstein, Fabio Scarabotti and Filippo Tolli is great and gives insight to the works of Persi Diaconis.
-The applications of the representation theory of the symmetric group in Hurwitz theory, see for example the book Riemann Surfaces and Algebraic Curves: A First Course in Hurwitz Theory by Renzo Cavalieri and Eric Miles .
For cluster algebras there is a forthcoming book by Sergey Fomin, Lauren Williams and Andrei Zelevinsky : https://arxiv.org/abs/1608.05735 .