A survey on spherical designs and algebraic combinatorics on spheres

Abstract

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 S n − 1 in R n . After discussing some important examples, we focus on tight spherical t -designs on S n − 1 . Tight t -designs have good combinatorial properties, but they rarely exist. So, we are interested in the finite subsets on S n − 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.