Let be a probability and let be the probability distribution on defined by
This exercise asks us to find the Fourier transform of with respect to the three different irreducible representations of . The carrot is that we’ll ‘learn something’.
If is the trivial representation defined by for each then since , we have . Clearly this holds whenever is a probability distribution.
Now suppose that is the sign representation of . So and . We have
Correspondingly, if then is unbiased with respect to odd and even permutations, while if then this Fourier coefficient detects the bias of towards one type of permutation.
Finally suppose that is the two-dimensional irreducible representation of . From the matrices in the book we see that
One interesting feature is that the contribution from the transpositions vanished. This can be foreseen: will commute with all the matrices for because it is a sum over a conjugacy class of . So by Schur’s lemma, it must be a scalar multiple of the identity matrix. But the trace of the matrix 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 cards in which two cards are chosen at random and then swapped. The associated probability distribution gives equal probability to all transpositions in . So we know without any calculation that there exists such that
Here denote the identity matrix of dimension .