关于从群轨道得到的求积公式和设计的注记
Note on cubature formulae and designs obtained from group orbits
AI总结：
本文基于Sobolev定理，给出Xu定理的简化证明，并将Neumaier-Seidel定理推广到不变欧几里得设计，从而分类由角向量轨道并集得到的紧欧几里得设计。
摘要：
1960年，Sobolev证明了对于有限反射群G，一个G不变的求积公式是t次的当且仅当它对所有次数不超过t的G不变多项式精确成立。本文发现了与Sobolev定理相关的不变求积公式和欧几里得设计的一些观察。首先，我们给出了Xu（1998）关于具有某种强对称性的求积公式存在性的充要条件定理的另一种证明。与Xu的原始证明相比，新证明更短更简单，并且给出了Xu定理中解析表达条件的普遍解释。其次，我们将Neumaier和Seidel（1988）关于欧几里得设计的定理推广到不变欧几里得设计，从而分类了由角向量轨道并集得到的紧欧几里得设计。这一结果将Bajnok（2007）关于在B型Weyl群下不变的紧欧几里得设计的分类定理推广到其他有限反射群。
英文摘要：
In 1960, Sobolev proved that for a finite reflection group G, a G-invariant cubature formula is of degree t if and only if it is exact for all G-invariant polynomials of degree at most t. In this paper, we find some observations on invariant cubature formulas and Euclidean designs in connection with the Sobolev theorem. First, we give an alternative proof of theorems by Xu (1998) on necessary and sufficient conditions for the existence of cubature formulas with some strong symmetry. The new proof is shorter and simpler compared to the original one by Xu, and moreover gives a general interpretation of the analytically-written conditions of Xu's theorems. Second, we extend a theorem by Neumaier and Seidel (1988) on Euclidean designs to invariant Euclidean designs, and thereby classify tight Euclidean designs obtained from unions of the orbits of the corner vectors. This result generalizes a theorem of Bajnok (2007) which classifies tight Euclidean designs invariant under the Weyl group of type B to other finite reflection groups.
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