🤖 AI Summary
This work proposes a unified framework for efficiently computing matrix elements—also known as group functions—of irreducible representations of the unitary group $U(d)$, accommodating both symbolic and numerical computation requirements. Built upon the Gelfand–Tsetlin basis, the approach integrates group representation theory, algorithms for Schur functions, and parameterizations of unitary matrices commonly used in quantum optics. It enables full representation construction, conversion between Gelfand–Tsetlin patterns and occupation-number states, and evaluation of Schur functions. The study presents the first general-purpose, high-performance library for $U(d)$ group functions implemented in Julia, offering seamless interoperability between quantum information science and representation theory, with export capabilities to Mathematica. Notably, the Wigner D-functions for $SU(2)$ emerge as a special case, significantly enhancing computational efficiency and cross-disciplinary compatibility.
📝 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.