Fourier inversion for finite inverse semigroups

TL;DR

This paper extends Fourier inversion theorems and develops fast inverse Fourier transforms for various classes of finite inverse semigroups, broadening the computational tools available for these algebraic structures.

Contribution

It introduces four inverse semigroup generalizations of the Fourier inversion theorem and constructs efficient FFTs for new classes of finite inverse semigroups.

Findings

Fast inverse Fourier transforms for the symmetric inverse monoid and wreath products.

FFT algorithms for the planar rook monoid, partial cyclic shift monoid, and partial rotation monoid.

General approach to constructing inverse Fourier transforms for finite inverse semigroups.

Abstract

This paper continues the study of Fourier transforms on finite inverse semigroups, with a focus on Fourier inversion theorems and FFTs for new classes of inverse semigroups. We begin by introducing four inverse semigroup generalizations of the Fourier inversion theorem for finite groups. Next, we describe a general approach to the construction of fast inverse Fourier transforms for finite inverse semigroups complementary to an approach to FFTs given in previous work. Finally, we give fast inverse Fourier transforms for the symmetric inverse monoid and its wreath product by arbitrary finite groups, as well as fast Fourier and inverse Fourier transforms for the planar rook monoid, the partial cyclic shift monoid, and the partial rotation monoid.