A survey on spherical designs and algebraic combinatorics on spheres

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 tt-designs on Sn−1Sn−1 in RnRn. After discussing some important examples, we focus on tight spherical tt-designs on Sn−1Sn−1. Tight tt-designs have good combinatorial properties, but they rarely exist. So, we are interested in the finite subsets on Sn−1Sn−1, which have properties similar to tight tt-designs from the various viewpoints of algebraic combinatorics. For example, rigid tt-designs, universally optimal tt-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 tt-designs, as well as general spherical tt-designs, and their constructions. We will discuss the relations between spherical tt-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)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 ττ function. Spherical tt-designs and Euclidean tt-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 tt-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.