Fast Fourier Transforms for the Rook Monoid

TL;DR

This paper introduces the Fourier transform for the rook monoid and presents the first efficient FFT algorithms for non-group semigroups, expanding the scope of Fourier analysis in algebraic structures.

Contribution

It defines the Fourier transform for the rook monoid and develops the first divide-and-conquer FFT algorithms for non-group semigroups.

Findings

Developed two efficient FFT algorithms for the rook monoid

Extended group FFT methods to non-group semigroups

First such extension in algebraic Fourier analysis

Abstract

We define the notion of the Fourier transform for the rook monoid (also called the symmetric inverse semigroup) and provide two efficient divide-and-conquer algorithms (fast Fourier transforms, or FFTs) for computing it. This paper marks the first extension of group FFTs to non-group semigroups.