GroupFunctions.jl: computing individual entries of the irreducible representations of the unitary group U(d)
2026-07-12 • Mathematical Software
Mathematical Software
AI summaryⓘ
The authors developed GroupFunctions.jl, a tool in the Julia programming language, to calculate special building blocks called matrix elements for certain mathematical groups related to symmetries. These matrix elements, known as group functions, can be calculated either exactly with symbols or approximately with numbers. For the simpler group SU(2), their tool matches well-known functions called Wigner D-functions. The library also helps translate between different mathematical bases and formats used in physics and quantum computing, and it can share results with software like Mathematica.
Juliamatrix elementsirreducible representationsU(d)SU(2)Wigner D-functionsGelfand-Tsetlin patternsSchur functionsquantum opticsMathematica
Authors
David Amaro-Alcalá, Konrad Szymański
Abstract
GroupFunctions.jl is a Julia library for computing individual matrix elements of irreducible representations of U(d). These matrix elements, called group functions, can be evaluated symbolically or numerically. For SU(2), they reduce to the Wigner D-functions. The library computes these matrix elements in a carrier-space basis enumerated by Gelfand-Tsetlin patterns. It can also compute entire representation operators, construct input unitaries from parameterisations common in quantum optics, translate Gelfand-Tsetlin patterns into occupation-number kets, and compute the associated Schur functions. Results can be exported in a form compatible with Mathematica.