Linear Time Fourier Transforms of Sn-k-invariant Functions on the Symmetric Group Sn

This paper introduces new techniques for the efficient computation of discrete Fourier transforms (DFTs) of Sn-k-invariant functions on the symmetric group S-n. We uncover diamond- and leaf-rake-like structures in Young's seminormal and orthogonal representations. Combining this with both a multiresolution scheme and an anticipation technique for saving scalar multiplications leads to linear time partial FFTs. Following the inductive version of Young's branching rule we obtain a global FF1 that computes a DFT Sn-k-invariant functions on S-n, in at most c(k). [S-n : Sn-k] scalar multiplications and additions, where c(k) denotes a positive constant depending only on k. This run-time, which is linear in [S-n : Sn-k], is order optimal and improves Maslen's algorithm. For example, it takes less than one second on a standard notebook to run our FFT algorithm for an Sn-2-invariant real-valued function on S-n, n = 5000.