Association schemes on general measure spaces and zero-dimensional Abelian groups

Abstract

Association schemes form one of the main objects of algebraic combinatorics, classically defined on finite sets. In this paper we define association schemes on arbitrary, possibly uncountable sets with a measure. We study operator realizations of the adjacency algebras of schemes and derive simple properties of these algebras. To develop a theory of general association schemes, we focus on schemes on topological Abelian groups where we can employ duality theory and the machinery of harmonic analysis. We construct translation association schemes on such groups using the language of spectrally dual partitions. Such partitions are shown to arise naturally on topological zero-dimensional Abelian groups, for instance, Cantor-type groups or the groups of p-adic numbers. This enables us to construct large classes of dual pairs of association schemes on zero-dimensional groups with respect to their Haar measure, and to compute their eigenvalues and intersection numbers. We also derive properties of infinite metric schemes, connecting them with the properties of the non-Archimedean metric on the group. Pursuing the connection between schemes on zero-dimensional groups and harmonic analysis, we show that the eigenvalues have a natural interpretation in terms of Littlewood-Paley wavelet bases, and in the (equivalent) language of martingale theory. For a class of nonmetric schemes constructed in the paper, the eigenvalues coincide with values of orthogonal functions on zero-dimensional groups. We observe that these functions, which we call Haar-like bases, have the properties of wavelets on the group, including in some special cases the self-similarity property. This establishes a seemingly new link between algebraic combinatorics and harmonic analysis. We conclude the paper by studying some analogs of problems of classical coding theory related to the theory of association schemes.