Applications of representation theory of the symmetric group Inspired by this question and in particular the final paragraph of Or's answer, I have the following question:


  Do you know of any applications of the representation theory of the symmetric group in TCS?


The symmetric group $S_n$ is the group of all permutations of $\{1, \ldots, n\}$ with group operation composition. A representation of $S_n$ is a homomorphism from $S_n$ to the general linear group of invertible $n \times n$ complex matrices. A representation acts on $\mathbb{C}^n$ by matrix multiplication. An irreducible representation of $S_n$ is an action that leaves no proper subspace of $\mathbb{C}^n$ invariant. Irreducible representations of finite groups allow one to define a Fourier transform over non-abelian groups. This Fourier transform shares some of the nice properties of the discrete Fourier transform over cyclic/abelian groups. For example convolution becomes pointwise multiplication in the Fourier basis.

The representation theory of the symmetric group is beautifully combinatorial. Each irreducible representation of $S_n$ corresponds to an integer partition of $n$. Has this structure and/or the Fourier transform over the symmetric group found any application in TCS?
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.