超八面体群的轨道作为欧几里得设计
Orbits of the hyperoctahedral group as Euclidean designs
AI总结：
本文研究由超八面体群轨道支撑的欧几里得设计，证明其强度只能为3、5或7，给出强度为5和7的充分必要条件，并构造了三个新的反极紧欧几里得设计。
摘要：
n维超八面体群H（李型B_n的Weyl群）是正交群中由所有坐标置换和关于坐标超平面的反射生成的子群。一个有限集X ⊂ R^n连同权函数w: X → R+被称为欧几里得t-设计，如果对每个全次数至多t的多项式f，等式∑_{r∈R} W_r f̄_{S_r} = ∑_{x∈X} w(x) f(x)成立；这里R是X中点的范数集合，W_r是X中范数为r的所有元素的总权，S_r是以原点为中心、半径为r的n维球面，f̄_{S_r}是f在S_r上的平均值。本文考虑由超八面体群的轨道支撑的欧几里得设计。具体地，我们证明广义超八面体并集上的任何欧几里得设计的强度（使其成为欧几里得设计的最大t值）等于3、5或7。我们找到了该强度为5和为7时的显式充分必要条件。为了建立我们的分类，我们将上述欧几里得设计的定义转化为t=5时的一个方程、t=7时的一组三个方程以及t=9时的一组七个方程。Neumaier和Seidel（1988）以及Delsarte和Seidel（1989）证明了关于R^n中p=|R|个同心球面上欧几里得t-设计最小规模的Fisher型不等式|X| ≥ N(n,p,t)（假设当t为奇数时设计是反极的）。恰好具有N(n,p,t)个点的欧几里得设计称为紧设计。我们给出了由超八面体群的轨道支撑的反极紧欧几里得设计的新例子，对应于N(n,p,t)=(3,2,5)、(3,3,7)和(4,2,7)。
英文摘要：
The hyperoctahedral group $H$ in $n$ dimensions (the Weyl group of Lie type $B_n$) is the subgroup of the orthogonal group generated by all transpositions of coordinates and reflections with respect to coordinate hyperplanes. A finite set ${\cal X} \subset \mathbb{R}^n$ with a weight function $w: {\cal X} \rightarrow \mathbb{R}^+$ is called a Euclidean $t$-design, if $$\sum_{r \in R} W_r \overline{f}_{S_{r}} = \sum_{{\bf x} \in {\cal X}} w({\bf x}) f({\bf x})$$ holds for every polynomial $f$ of total degree at most $t$; here $R$ is the set of norms of the points in ${\cal X}$, $W_r$ is the total weight of all elements of ${\cal X}$ with norm $r$, $S_r$ is the $n$-dimensional sphere of radius $r$ centered at the origin, and $\overline{f}_{S_{r}}$ is the average of $f$ over $S_{r}$. Here we consider Euclidean designs which are supported by orbits of the hyperoctahedral group. Namely, we prove that any Euclidean design on a union of generalized hyperoctahedra has strength (maximum $t$ for which it is a Euclidean design) equal to 3, 5, or 7. We find explicit necessary and sufficient conditions for when this strength is 5 and for when it is 7. In order to establish our classification, we translate the above definition of Euclidean designs to a single equation for $t=5$, a set of three equations for $t=7$, and a set of seven equations for $t=9$. Neumaier and Seidel (1988), as well as Delsarte and Seidel (1989), proved a Fisher-type inequality $|{\cal X}| \geq N(n,p,t)$ for the minimum size of a Euclidean $t$-design in $\mathbb{R}^n$ on $p=|R|$ concentric spheres (assuming that the design is antipodal if $t$ is odd). A Euclidean design with exactly $N(n,p,t)$ points is called tight. We exhibit new examples of antipodal tight Euclidean designs, supported by orbits of the hyperoctahedral group, for $N(n,p,t)=$(3,2,5), (3,3,7), and (4,2,7).
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