GroupFFT.jl

Overview

• A Julia package for performing fast Fourier transforms over arbitrary compact groups

• Currently supports the rotation group, SO(3), and the rotation/reflection group, O(3).

• Licensed under the MIT License

How to install

In Julia:

Pkg.clone("https://github.com/NickMcNutt/GroupFFT.jl")

Examples

Generate an element of the orthogonal group O(3) using the Euler ZYZ parameterization:

using GroupFFT

α, β, γ = π/2, π/2, π/2
g = O3(α, β, γ)

Compute the unitary irreducible representations of element g. For the group O(3), these are Wigner-D matrices.

unitary_irreps = [unitary_irrep(g, i) for i in 0:2]
display.(round.(unitary_irreps, 3))

1×1 Array{Complex{Float64},2}:
 1.0+0.0im

3×3 Array{Complex{Float64},2}:
 -0.5+0.0im     0.0+0.707im   0.5+0.0im  
 -0.0-0.707im   0.0+0.0im     0.0-0.707im
  0.5+0.0im    -0.0+0.707im  -0.5-0.0im  

5×5 Array{Complex{Float64},2}:
   0.25-0.0im  -0.0-0.5im  -0.612+0.0im  0.0+0.5im    0.25+0.0im
    0.0+0.5im   0.5-0.0im     0.0+0.0im  0.5+0.0im     0.0-0.5im
 -0.612+0.0im  -0.0-0.0im    -0.5-0.0im  0.0-0.0im  -0.612-0.0im
   -0.0-0.5im   0.5+0.0im    -0.0+0.0im  0.5+0.0im    -0.0+0.5im
   0.25+0.0im  -0.0+0.5im  -0.612-0.0im  0.0-0.5im    0.25+0.0im

If we don't want complex numbers, we can generate real irreducible representations instead:

orthogonal_irreps = [orthogonal_irrep(g, i) for i in 0:2]
display.(round.(orthogonal_irreps, 3))

1×1 Array{Float64,2}:
 1.0

3×3 Array{Float64,2}:
 -1.0  -0.0  0.0
  0.0  -0.0  1.0
 -0.0   1.0  0.0

5×5 Array{Float64,2}:
 -0.0  -1.0  -0.0     0.0  -0.0  
 -1.0   0.0   0.0    -0.0   0.0  
  0.0  -0.0  -0.5    -0.0   0.866
 -0.0   0.0  -0.0     1.0   0.0  
  0.0  -0.0   0.866   0.0   0.5

Functionality

GroupFFT.jl specifies an abstract base type Group from which all group types derive. Currently, support is available for the following groups:

OrthogonalGroup{T, 3}
SpecialOrthogonalGroup{T, 3}
SymmetricGroup{T, N}

For convenience, type aliases are provided for groups of low dimension:

O3{T}
SO3{T}

Contributing

GroupFFT.jl aims to be a comprehensive package that provides base types and linear representations for all of the most widely used groups. We welcome the support of anyone who wishes to contribute to this project.

Completed

• Base types and irreducible representations for O(3), SO(3), and S_n

Todo

• Base types and representations for O(n), U(n), SU(n), SL(n), SE(n), PSL(n)
• Support for other kinds of group representations (standard, faithful, etc.)
• Support for fields other than the complex and real numbers