Separation of Variables and the Computation of Fourier Transforms on Finite Groups, II

Abstract

We present a general diagrammatic approach to the construction of efficient algorithms for computing the Fourier transform of a function on a finite group. By extending work which connects Bratteli diagrams to the construction of Fast Fourier Transform algorithms we make explicit use of the path algebra connection to the construction of Gel'fand-Tsetlin bases and work in the setting of quivers. We relate this framework to the construction of a configuration space derived from a Bratteli diagram. In this setting the complexity of an algorithm for computing a Fourier transform reduces to the calculation of the dimension of the associated configuration space. Our methods give improved upper bounds for computing the Fourier transform for the general linear groups over finite fields, the classical Weyl groups, and homogeneous spaces of finite groups, while also recovering the best known algorithms for the symmetric group and compact Lie groups.Date: December 9, 2015. 2000 Mathematics Subject Classification. To be filled in.Definition 2.7. A quiver Q is a directed multigraph with vertex set V (Q) and edge set E(Q). For an arrow (directed edge) e ∈ E(Q) from vertex β to vertex α, we call α the target of e and β the source of e.Let Q be a quiver. For each e ∈ E(Q), let t(e) denote the target of e and s(e) the source of e. Definition 2.8. A quiver Q is graded if there is a function gr : V (Q) → N such that for each e ∈ E(Q), gr(t(e)) > gr(s(e)).Example 2.9. Figure 1 is an example of a graded quiver. Each vertex v is labeled by its grading, gr(v).

Fast Generalized DFTs for all Finite Groups

“…In the 1990s, Maslen, Rockmore, and coauthors developed the so-called "separation of variables" approach [MR97a], which relies on non-trivial decompositions along chains of subgroups via Bratteli diagrams and detailed knowledge of the representation theory of the underlying groups. There is a rather large body of literature on this approach and it has been applied to a wide variety of group algebras and more general algebraic objects.…”

A fast generalized DFT for finite groups of Lie type

“…There is a rather large body of literature on this approach and it has been applied to a wide variety of group algebras and more general algebraic objects. For a fuller description of this approach and the results obtained, the reader is referred to the surveys [MR97b, MR97a,Roc02], and the most recent paper in this line of work [MRW16a].…”

“…Theorem 5.14 in [MR97a] Figure 1: Previously best known running times for the generalized DFT over various families of linear groups. In this table, the O(·) notation hides lower order terms and the dependence on n.…”

Recent Progress and Applications in Group FFTs

“…[29,36] for the case of the symmetric group). The full formalism [29,34] phrases all of this in the language of bilinear maps and bears some resemblance to the fundamental FFT work of Winograd [56].…”

“…These are among the classes of groups for which O|G| log c |G| Fourier transform algorithms are known. Other examples include the supersolvable groups [3], while the algorithms for finite matrix groups and Lie groups of finite type still have room for improvement [33,36].…”