GroupFunctions.jl

A Julia library to compute D-functions, which are entries of the irreducible representations of the unitary group U(d). These entries can be numeric or symbolic.

Highlights

• Numerical and symbolic D-functions for U(d) irreps
• Character computation for SU(d) from traces of representation matrices
• Gelfand–Tsetlin basis construction via basis_states
• Works with Haar-random unitaries (e.g. using RandomMatrices.jl)
• Export symbolic results to Mathematica with julia_to_mma

Installation

Requires Julia 1.6 or newer.

• From the Julia registry (recommended):

pkg> add GroupFunctions

• From this repository:

user@machine:~$ mkdir new_code && cd new_code
user@machine:~$ julia --project=.
julia> ] add https://github.com/davidamaro/GroupFunctions.jl

Installing Julia

• Mac: Use juliaup. Installing Julia via brew is not recommended.
• Linux: Use the appropriate package manager (e.g., sudo pacman -S julia).
• Windows: Run winget install julia -s msstore in your terminal and follow the steps.

Quick start

using GroupFunctions
using RandomMatrices  # Optional: for Haar-random unitaries
using LinearAlgebra: det

irrep = [2, 1, 0]
U = rand(Haar(2), 3)  # 3×3 Haar-random unitary matrix
basis = basis_states(irrep)

# Numerical D-function entry
group_function(irrep, basis[1], basis[3], U)

# SU(d) character
U_su = U / det(U)^(1/size(U, 1))  # enforce det=1 for SU(d)
character(irrep, U_su)

# Symbolic D-function entry
sym = group_function(irrep, basis[1], basis[3])
julia_to_mma(sym)  # Mathematica-friendly expression

For more examples and API details, see the documentation: https://davidamaro.github.io/GroupFunctions.jl/dev/

Contributing

Contributions are welcome. Please open an issue before writing code, and read the contribution guidelines first. Questions and bug reports belong in the issue tracker too.

For local development convenience:

• Run tests: julia --project -e 'using Pkg; Pkg.test()'
• Build docs locally: julia --project=docs docs/make.jl

License

GroupFunctions.jl is distributed under the MIT License (see LICENSE).

References

1. J Grabmeier and A Kerber, "The evaluation of irreducible polynomial representations of the general linear groups and of the unitary groups over fields of characteristic 0" Acta Appl. Math, 1987
2. A Alex et al, "A numerical algorithm for the explicit calculation of SU(N) and SL(N, C) Clebsch–Gordan coefficients" J. Math. Phys. 2011
3. D Amaro-Alcala et al "Sum rules in multiphoton coincidence rates" Phys. Lett. A 2020

Citation

Citation information is pending. In the meantime, please cite the repository URL and version tag (e.g., “GroupFunctions.jl v0.3.1, 2026, https://github.com/davidamaro/GroupFunctions.jl”).