Random walks on the symmetric group generated by conjugacy classes

The object of this study is conjugacy-invariant random walks on the symmetric group. We determine the rates of convergence for a number of generating conjugacy classes, compare mixing speeds of different classes, and establish monotonicity relations for the distributions arising from the walks. The primary tool of this analysis is the non-commutative Fourier transform, which translates questions about random walks into problems about characters of the group. We use combinatorial techniques involving rim hook tableaux to obtain several results about representations of the symmetric and alternating groups, including an upper bound for a family of characters and an asymptotics for the sum of inverted dimensions.