On the Fourier transform for a symmetric group homogeneous space

TL;DR

This paper derives a simplified Fourier transform formula for permutations acting on symmetric group homogeneous spaces, demonstrating its computational efficiency with a specific number of multiplications and additions.

Contribution

It introduces a straightforward form of the Fourier transform for symmetric group homogeneous spaces using Young orthogonal representations, highlighting its computational advantages.

Findings

Transform requires 2n-2 multiplications and additions

Simplifies Fourier analysis on symmetric group spaces

Provides a computationally efficient method

Abstract

By using properties of the Young orthogonal representation, this paper derives a simple form for the Fourier transform of permutations acting on the homogeneous space of $n$-dimensional vectors, and shows that the transform requires $2n-2$ multiplications and the same number of additions.