## 2B Exercise 2: Fourier transform example

Let $p$ be a probability and let $P$ be the probability distribution on $S_3$ defined by

$P(1) = p, P(12) = P(13) = P(23) = (1-p)/3.$

This exercise asks us to find the Fourier transform of $P$ with respect to the three different irreducible representations of $S_3$. The carrot is that we’ll ‘learn something’.

If $\rho : S_3 \rightarrow \mathbf{C}$ is the trivial representation defined by $\rho(s) = 1$ for each $s \in S_3$ then since $\sum_{s \in S_3} P(s) = 1$, we have $\hat{P}(\rho) = 1$. Clearly this holds whenever $P$ is a probability distribution.

Now suppose that $\rho$ is the sign representation of $G$. So $\rho(\mathrm{id}) = \rho(123) = \rho(132) = 1$ and $\rho(12) = \rho(23) = \rho(13) = -1$. We have

$\hat{P}(\rho) = p - 3(1-p)/3 = 2p - 1.$

Correspondingly, if $p = 1/2$ then $P$ is unbiased with respect to odd and even permutations, while if $p \not= 1/2$ then this Fourier coefficient detects the bias of $P$ towards one type of permutation.

Finally suppose that $\rho$ is the two-dimensional irreducible representation of $S_3$. From the matrices in the book we see that

$\rho(12) + \rho(23) + \rho(13) = \left( \begin{matrix} 0 & 0 \\ 0 & 0 \end{matrix} \right)$

so

$\hat{P}(\rho) = p \left( \begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix} \right).$

One interesting feature is that the contribution from the transpositions vanished. This can be foreseen: $\rho(12) + \rho(13) + \rho(23)$ will commute with all the matrices $\rho(s)$ for $s \in S_3$ because it is a sum over a conjugacy class of $S_3$. So by Schur’s lemma, it must be a scalar multiple of the identity matrix. But the trace of the matrix $\rho(12)$ is zero, so this multiple must be zero. (Something similar happens with the Casimir element in representations of semi-simple Lie algebras.)

This shortcut might be useful in bigger examples: for example, consider the shuffle of $n$ cards in which two cards are chosen at random and then swapped. The associated probability distribution $P$ gives equal probability to all transpositions in $S_n$. So we know without any calculation that there exists $\lambda_\rho \in \mathbf{C}$ such that

$\hat{P}(\rho) = \lambda_\rho \mathbf{1}.$

Here $\mathbf{1}$ denote the identity matrix of dimension $\mathrm{dim}\ \rho$.