GroupFunctions.jl: computing individual entries of the irreducible representations of the unitary group U(d)

Jul 16, 2026

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GroupFunctions.jl is a Julia library for computing individual matrix elements of irreducible representations of the unitary group U(d), termed group functions. It evaluates these elements symbolically or numerically and uses a carrier-space basis indexed by Gelfand–Tsetlin patterns. For SU(2), results reduce to Wigner D-functions. The library also builds full representation operators, constructs input unitaries from quantum-optics parameterisations, converts Gelfand–Tsetlin patterns into occupation-number kets, computes Schur functions, and exports formats compatible with Mathematica.

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Abstract

Statement of Need

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Example Use and Documentation

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arXiv :2607 . 10867v1 [ quant-ph] 12 Jul 2026 GroupFunctions.jl: computing individual entries of the irreducible representations of the unitary group U(d) David Amaro-Alcalá∗ and Konrad Szymański† Research Centre for Quantum Information, Institute of Physics, Slovak Academy of Sciences, Dúbravská cesta 9, Bratislava, Slovakia (Dated: 2026-07-07) Abstract GroupFunctions.jla 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. I. STATEMENT OF NEED Representations of the unitary group U (d) arise in many subfields of physics and mathematics, and computations often reduce to evaluating their matrix elements, called group functions. For an irrep labelled by λ, the corresponding object is Do()t ,init (U ) = ⟨out | D(λ)(U ) | init⟩ , where init and out denote basis states in the representation carrier space. In mathematics, summing the diagonal group functions gives the trace of the representation matrix of U. This trace is the character of the representation and the Schur polynomial of the eigenvalues of U , an object central to algebraic combinatorics and symmetric function theory. In quantum physics, group functions appear in several settings. In quantum optics, a group function gives the transition amplitude of photons through a linear optical network. The same object can be used to characterise quantum devices [1] and to describe the symmetry properties of states [2] . One important subproblem is boson sampling [3], where the transition amplitude reduces to a permanent whose evaluation is classically computationally hard. Whether noisy real-world quantum devices can perform this computation remains an active question. Some of these tasks require only a numerical estimate, whereas others require an exact symbolic group function. GroupFunctions.jl addresses the latter need. Although related packages exist (see below), to our knowledge, none is designed to compute individual representation-matrix entries symbolically. When applicable, computing an individual entry is more eﬀicient than assembling the entire matrix. a David Amaro-Alcalá developed the package, its algorithms, and its test suite, and prepared the initial documentation. Konrad Szymański substantially revised and expanded the documentation and developed additional examples, and contributed to the standardisation of the API. ∗ ORCID: 0000-0001-8137-2161 † ORCID: 0000-0001-7676-1605 A. Similar software Several existing packages relate to GroupFunctions.jl but address different problems. SUNRepresentations.jl [4] and the algorithm of Alex et al. [5] (with appendix code) compute SU (d) Clebsch-Gordan coeﬀicients. RepLAB [6] supports manipulating irreducible representations of various groups, including U (d), but provides only indirect numerical access to group functions. IntegrateUnitary.jl [7] performs symbolic integration over compact groups rather than evaluating representation matrices. haarpy [8] implements Weingartencalculus methods. Other libraries focus on quantum optics. BosonSampling.jl [9], Perceval [10], and QOptCraft [11] numerically model linear optical devices, while The Walrus [12] helps compute amplitudes for Gaussian boson sampling. These packages do not target the symbolic computation of individual representation-matrix entries, which is the primary purpose of GroupFunctions.jl. II. EXAMPLE USE AND DOCUMENTATION The library pr

FAQs

What does GroupFunctions.jl compute for the unitary group U(d)?

It computes individual matrix elements of irreducible representations of U(d), called group functions, between specified basis states in the representation carrier space.

How are group functions represented and evaluated in this library?

Group functions are computed in a carrier-space basis enumerated by Gelfand–Tsetlin patterns, and can be evaluated symbolically or numerically; for SU(2) they reduce to Wigner D-functions.

What additional capabilities does the library provide beyond single matrix entries?

It can compute entire representation operators, construct input unitaries from parameterisations common in quantum optics, translate Gelfand–Tsetlin patterns into occupation-number kets, and compute associated Schur functions, with export support for Mathematica.

Jul 16, 2026

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