Fixed points of non-uniform permutations and representation theory of the symmetric group

We use representation theory of the symmetric group S_n to prove Poisson limit theorems for the distribution of fixed points for three types of non-uniform permutations. First, we give results for the commutator of x and g where g and x are uniform in S_n. Second, we give results for the commutator of x and g where g is uniform in S_n and x is fixed. Third, we give results for permutations obtained by multiplying n*log(n)/2 + cn random transpositions, as well as a conjecture related to the more general i-cycle walk. Most of our results are known by other, quite different, methods.