Discrete orthogonality relations for Hall–Littlewood polynomials are employed so as to derive cubature rules for the integration of homogeneous symmetric functions with respect to the density of the circular unitary ensemble (which originates from the Haar measure on the special unitary group |$SU(n;\mathbb{C})$|⁠). By passing to Macdonald’s hyperoctahedral Hall–Littlewood polynomials, we moreover find analogous cubature rules for the integration with respect to the density of the circular quaternion ensemble (which originates in turn from the Haar measure on the compact symplectic group |$Sp (n;\mathbb{H})$|⁠). The cubature formulas under consideration are exact for a class of rational symmetric functions with simple poles supported on a prescribed complex hyperplane arrangement. In the planar situations (corresponding to |$SU(3;\mathbb{C})$| and |$Sp (2;\mathbb{H})$|⁠), a determinantal expression for the Christoffel weights enables us to write down compact cubature rules for the integration over the equilateral triangle and the isosceles right triangle, respectively.

中文翻译：

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Hall–Littlewood多项式的古巴规则

采用霍尔-利特伍德多项式的离散正交关系，以得出关于圆形unit系综密度的均质对称函数积分的库克斯规则（该规则源自于特殊|群| $ SU（n ; \ mathbb {C}）$ |⁠）。通过传递给麦克唐纳德的超八面体霍尔–利特伍德多项式，我们还找到了关于圆形四元数集合体密度的积分的类似空间规则（这又源于紧凑辛群| $ Sp（n; \ mathbb {H}）$ |⁠）。所考虑的容积公式对于一类有理对称函数具有简单的极点，并在规定的复杂超平面结构上提供了支持。在平面情况下（对应于| $ SU（3; \ mathbb {C}）$ |和| $ Sp（2; \ mathbb {H}）$ |⁠），Christoffel权重的行列式使我们能够编写向下紧凑的培养空间规则，分别在等边三角形和等腰直角三角形上进行积分。