Fast Fourier transforms for symmetric groups: theory and implementation

Abstract

Recently, it has been proved that a Fourier transform for the symmetric group S n {S_n} based on Young’s seminormal form can be evaluated in less than 0.5 ( n 3 + n 2 ) n ! 0.5({n^3} + {n^2})n! arithmetic operations. We look at this algorithm in more detail and show that it allows an efficient software implementation using appropriate data structures. We also describe a similarly efficient algorithm for the inverse Fourier transform. We compare the time and memory requirements of our program to those of other existing implementations.