构造两个低阶球面设计的并的高阶球面设计
Constructing high order spherical designs as a union of two of lower order
摘要：
我们展示了如何变分表征的球形设计可以用来采取一个联盟的球形设计，以获得一个球形设计的高阶（度，精度，精确度）与少量的点。我们考虑的例子涉及采取的两个向量的轨道下的一个复杂的反射群的作用，以获得一个加权的球形$(t,t)$设计。这些设计具有高度的对称性（与点的数量相比），并且许多是这种设计的第一已知构造，例如，对于$\mathbb{C}^2$的$32$点$(9,9)$ -设计，对于$\mathbb{C}^3$的$48$点$(4,4)$ -设计，以及对于$\mathbb{C}^4$的$400$点$(5,5)$ -设计。从实反射群，我们构造对于$\mathbb{R}^4$的$360$点$(9,9)$ -设计（$18$阶的球面半设计），即，一种$720$点球型$19$ -为$\mathbb{R}^4$设计。
英文摘要：
We show how the variational characterisation of spherical designs can be used to take a union of spherical designs to obtain a spherical design of higher order (degree, precision, exactness) with a small number of points. The examples that we consider involve taking the orbits of two vectors under the action of a complex reflection group to obtain a weighted spherical $(t,t)$-design. These designs have a high degree of symmetry (compared to the number of points), and many are the first known construction of such a design, e.g., a $32$ point $(9,9)$-design for $\mathbb{C}^2$, a $48$ point $(4,4)$-design for $\mathbb{C}^3$, and a $400$ point $(5,5)$-design for $\mathbb{C}^4$.From a real reflection group, we construct a $360$ point $(9,9)$-design for $\mathbb{R}^4$ (spherical half-design of order $18$), i.e., a $720$ point spherical $19$-design for $\mathbb{R}^4$.
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