A survey on spherical designs and algebraic combinatorics on spheres
A survey on spherical designs and algebraic combinatorics on spheres
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DOI：
10.1016/j.ejc.2008.11.007
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发表时间：
2009-08
期刊：
影响因子：
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通讯作者：
E. Bannai;E. Bannai
中科院分区：
文献类型：
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作者：
E. Bannai;E. Bannai
本调查主要针对非专业人士，但我们也试图包括专家可能感兴趣的许多最新发展。我们想研究单位球面的“好”有限子集。考虑“什么是好的”是我们问题的一部分。我们首先定义Sn− 1 in Rn上的球面t-设计。在讨论了一些重要的例子之后，我们将重点讨论Sn−1上的紧球面t-设计。紧t-设计具有良好的组合性质，但它们很少存在。因此，我们对Sn−1上的有限子集感兴趣，从代数组合学的各种观点来看，它们具有类似于紧t-设计的性质。例如，刚性t-设计，普遍最优的t-码（配置），以及有限集，其中承认的结构的关联计划，是其中之一。我们将讨论各种结果的存在性和不存在的特殊球形t-设计，以及一般的球形t-设计，和他们的建设。我们将讨论球面t-设计和许多其他数学分支之间的关系。举例来说：通过考虑作为真实的正交群O（n）中有限群的轨道的球面设计，我们得到了与群论的许多联系; 2通过考虑作为欧氏格壳的球面设计，我们得到了与数论的许多意想不到的联系，如模形式和关于Ramanujan τ函数零点的Lehmer猜想。球面t-设计和欧几里德t-设计是逼近论中求积公式的特殊情况，因此我们与分析和统计，特别是与正交多项式和矩问题有许多联系。此外，德尔萨特的线性规划方法和许多最近的推广，包括工作的穆辛和随后的进展，在使用半定规划，有很强的联系与几何（特别是球包装问题）和理论的优化。这些不同的联系解释了球面代数组合学的魅力的原因。与此同时，这些球面t-设计理论和相关主题在一般代数组合学的发展中有很强的根源，这是在结合方案的框架下作为Delsarte理论的代码和设计开始的。
This survey is mainly intended for non-specialists, though we try to include many recent developments that may interest the experts as well. We want to study “good” finite subsets of the unit sphere. To consider “what is good” is a part of our problem. We start with the definition of spherical t-designs on Sn−1in Rn. After discussing some important examples, we focus on tight spherical t-designs on Sn−1. Tight t-designs have good combinatorial properties, but they rarely exist. So, we are interested in the finite subsets on Sn−1, which have properties similar to tight t-designs from the various viewpoints of algebraic combinatorics. For example, rigid t-designs, universally optimal t-codes (configurations), as well as finite sets which admit the structure of an association scheme, are among them. We will discuss various results on the existence and the non-existence of special spherical t-designs, as well as general spherical t-designs, and their constructions. We will discuss the relations between spherical t-designs and many other branches of mathematics. For example: by considering the spherical designs which are orbits of a finite group in the real orthogonal group O(n), we get many connections with group theory; by considering those which are shells of Euclidean lattices, we get many unexpected connections with number theory, such as modular forms and Lehmer’s conjecture about the zeros of the Ramanujan τ function. Spherical t-designs and Euclidean t-designs are special cases of cubature formulas in approximation theory, and thus we get many connections with analysis and statistics, and in particular with orthogonal polynomials, and moment problems. Moreover, Delsarte’s linear programming method and many recent generalizations, including the work of Musin and the subsequent progress in using semi-definite programming, have strong connections with geometry (in particular sphere packing problems) and the theory of optimizations. These various connections explain the reason of the charm of algebraic combinatorics on spheres. At the same time, these theories of spherical t-designs and related topics have strong roots in the developments of algebraic combinatorics in general, which was started as Delsarte theory of codes and designs in the framework of association schemes.