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

We use representation theory of the symmetric group Sₙ 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 Sₙ. Second, we give results for the commutator of x and g where g is uniform in Sₙ 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.