Groups.jl

Overview

• A generic Julia package for working with groups

• Able to compute linear group representations (including irreducible representations)

• Code is focused on high performance and high generality

• Serves as a base library for other group-oriented packages, like GroupFFT.jl

• Licensed under the MIT License

How to install

In Julia:

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

Examples

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

using Groups

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

Groups.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

Groups.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