Fast Fourier Transforms for Finite Inverse Semigroups

TL;DR

This paper generalizes fast Fourier transform algorithms from finite groups to finite inverse semigroups, enabling efficient computation of Fourier transforms on these algebraic structures, including rook monoids and their wreath products.

Contribution

It introduces a general method for constructing irreducible representations of finite inverse semigroups and develops explicit fast algorithms for specific inverse semigroups.

Findings

Reduces Fourier transform computation to subgroup transforms and zeta transforms.

Provides explicit fast algorithms for rook monoids and wreath products.

Demonstrates efficiency improvements for Fourier transforms on inverse semigroups.

Abstract

We extend the theory of fast Fourier transforms on finite groups to finite inverse semigroups. We use a general method for constructing the irreducible representations of a finite inverse semigroup to reduce the problem of computing its Fourier transform to the problems of computing Fourier transforms on its maximal subgroups and a fast zeta transform on its poset structure. We then exhibit explicit fast algorithms for particular inverse semigroups of interest--specifically, for the rook monoid and its wreath products by arbitrary finite groups.