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.

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