Linear time Fourier transforms of Sn−k-invariant functions on the symmetric group Sn

Abstract

This paper introduces new techniques for the efficient computation of discrete Fourier transforms (DFTs) of Sn−k-invariant functions on the symmetric group Sn. We uncover diamond- and leaf-rake-like structures in Young's seminormal and orthogonal representations leading to relatively expensive diamond and cheaper leaf-rake computations. These local computations constitute the basis of a reduction/induction process. We introduce a new anticipation technique that avoids diamond computations at the expense of only a small arithmetic overhead for leaf-rake computations. This results in local fast Fourier transforms (FFTs). Combining these local FFTs with a multiresolution scheme close related to the inductive version of Young's branching rule we obtain a global FFT algorithm that computes the DFT of Sn−k-invariant functions on Sn in linear time. More precisely, we show that for fixed k and all n≥2k DFTs of Sn−k-invariant functions on Sn can be computed in at most ck⋅[Sn:Sn−k] scalar multiplications and additions, where ck denotes a positive constant depending only on k. This run-time is order-optimal and improves Maslen's algorithm.