Applications of representation theory of the symmetric group
Here are a few other examples. 


Diaconis and Shahshahani (1981) studied how many random transpositions are required in order to generate a near uniform permutation. They proved a sharp threshold of 1/2 n log(n) +/- O(n). Generating a Random Permutation with Random Transpositions.
Kassabov (2005) proved that one can build a bounded degree expander on the symmetric group. Symmetric Groups and Expander Graphs.
Kuperberg, Lovett and Peled (2012) proved that there exist small sets of permutations which act uniformly on k-tuples. Probabilistic existence of rigid combinatorial structures.