AI 中文总结
研究利用Julia库GroupFunctions.jl计算酉群U(d)不可约表示的单个矩阵元素，通过盖尔范德-采特林模式在载体空间基中计算，还具备多种功能，可导出与Mathematica兼容的结果。
AI 中文摘要
本文介绍了一个Julia库GroupFunctions.jl，用于计算U(d)不可约表示的单个矩阵元素，即群函数。这些矩阵元素可符号化或数值化求值，对于SU(2)，它们简化为维格纳D函数。该库在由盖尔范德-采特林模式枚举的载体空间基中计算这些矩阵元素，还能计算整个表示算子等。结果可导出为与Mathematica兼容的形式。
英文摘要
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.
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