Fast Fourier transforms for the rook monoid

Abstract

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

Inverse semigroup spectral analysis for partially ranked data

“…It is easy to see that this works regardless of the particular matrix representations chosen for Y. Computationally efficient methods for computing Fourier transforms and their inverses on a wide variety of groups and semigroups have been developed. See, for example, [1,3,5,11,12,13,14,15,21]. Example 3.6 (Symmetric group spectral analysis).…”

Fourier Inversion for Finite Inverse Semigroups

“…In [17] we developed an application of Fourier analysis on the rook monoid to the analysis of partially ranked datasets. While we proved a handful of Fourier inversion theorems for R n in [15], there has not yet been a treatment of Fourier inversion for arbitrary inverse semigroups, nor has there been a treatment of fast Fourier inversion for inverse semigroups. This paper addresses these issues.…”

“…Every group is an inverse semigroup, but not conversely. In [15,16] we extended the theory of Fourier analysis on finite groups to finite inverse semigroups and developed explicit FFTs for the symmetric inverse monoid (also called the rook monoid) R n and its wreath product by arbitrary finite groups.…”

“…Unlike the natural basis, Fourier bases of CS are defined by symmetry conditions on S (Definition 2.9). If S = Z n , the Fourier transform of f ∈ CZ n is the usual DFT of f , and the Fourier basis for CZ n is the usual basis of exponential functions [15,16,17].…”

The Efficient Computation of Fourier Transforms on Semisimple Algebras

“…The complexity of A, C(A) is the least upper bound of an algorithm effecting such a map and thus bounded above a priori by dim(A) 2 . The work presented in this paper, aiming to reduce this naive upper bound, is both motivated as a next "natural" step in algebraic FFT work (see also extensions to the semigroup case [29,27,28]) as well as by a particular application: the study of a certain random walk on the Birman-Murakami-Wenzl (BMW) algebra [47]. The usefulness of Fourier analysis for studying random walks on finite groups and algebras is well known (see e.g., [12,13]) and this paper thus connects with that literature as well.…”