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.

• Numerical and symbolic D-functions for U(d) irreps
• Character computation for SU(d) via the Weyl/Schur determinant formula
• 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

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

• 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.

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

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 via Weyl/Schur determinant formula
U_su = U / det(U)^(1/size(U, 1))  # enforce det=1 for SU(d)
character_weyl(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/

using GroupFunctions
using RandomMatrices
using LinearAlgebra: det
using Statistics: var

nsamples = 1_000
n = 3
irrep = [2, 1, 0]

values = Vector{ComplexF64}(undef, nsamples)
for i in 1:nsamples
    U = rand(Haar(2), n)
    U_su = U / det(U)^(1 / n)
    values[i] = character_weyl(irrep, U_su)
end

variance = var(values)
# Expect variance ≈ 1 for these samples.
variance

Contributions are welcome! Please feel free to submit a Pull Request. A to-do list is included in the todo.txt file.

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

GroupFunctions.jl is distributed under the MIT License (see LICENSE). While the package still contains code derived from AbstractAlgebra.jl to handle Young tableaux, it follows the same (MIT) license terms.

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
4. AbstractAlgebra.jl

Citation information is pending. In the meantime, please cite the repository URL and version tag (e.g., “GroupFunctions.jl v0.1.5, 2024, https://github.com/davidamaro/GroupFunctions.jl”).