rt.representation theory - Fun applications of representations of finite groups

Maybe this is not an "application", but I certainly found it fun when learning about representations of the symmetric group. Combinatorially, there is a clear correspondence between transpose-invariant partitions (ie partition diagrams that are symmetric along the main diagonal) and partitions involving distinct odd integers. (eg (3,2,1) corresponds to (5,1). )

Here is a sketch of the representation theory behind the result: the Specht modules from transpose-invariant partitions are precisely the irreducible representations of S_n that decompose into 2 irreducible representations when restricted to A_n. We view irreducible characters and conjugacy-class-indicator-functions as two bases on the vector space of A_n class functions, and deduce that the number of these reducible-on-restriction representations is equal to the number of S_n conjugacy classes which split as two A_n conjugacy classes. An S_n conjugacy class splits into two A_n conjugacy classes precisely when it doesn't commute with any odd cycles, which is to say all factors of its cycle decomposition have distinct odd length.

I've seen other instances where representation theory "explains" a combinatorial coincidence (eg q-dimension formula of various Lie algebras), so I think of this example as "typical" of the connection between representation theory and combinatorics.

EDIT: The background comes from pages 18-25 of the lecture notes here: http://www.dpmms.cam.ac.uk/~ae284/characters.html , and this particular statement is exercise 1 of sheet 3. I have what I believe is a complete solution to the exercise, but I'm having a little trouble pdf-ing it, hopefully it should be on my homepage (under "writings") by Tuesday (along with my own exposition of the background, which may expand on those official notes a little).