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?