What are some non trivial results that can be proved using representation theory that are interesting without a lot of technical representation theory knowledge? Let me give some examples to give you an idea of the kind of results I am looking for. For instance in algebraic topology quick consequences of the properties of the fundamental group are the fundamental theorem of algebra and brouwers fixed point theorem in 2d. Later on you can prove interesting results like the only finite dimensional commutative division algebras over Reals with identity are R and C, dimensional invariance and jordan curve theorem. You can also prove not so classical but still interesting results like S^n is a H space for n=0,1,3,7 this can be appreciated with little knowledge in homotopy theory. Or for instance complex analysis has the beautiful proof of the fundamental theorem of algebra or the analyticity of holomorphic functions.
I understand that it's possible that there aren't many such classical applications of representation theory as Gian Carlo Rota wrote
> 'What can you prove with exterior algebra that you cannot prove without it?' Whenever you hear this question raised about some new piece of mathematics, be assured that you are likely to be in the presence of something important. In my time, I have heard it repeated for random variables, Laurent Schwartz’ theory of distributions, ideles and Grothendieck’s schemes, to mention only a few. A proper retort might be: 'You are right. There is nothing in yesterday’s mathematics that could not also be proved without it. Exterior algebra is not meant to prove old facts, it is meant to disclose a new world. Disclosing new worlds is as worthwhile a mathematical enterprise as proving old conjectures.'
-- "Indiscrete Thoughts"
I am making this post to get some motivation to read representation theory.