On a Compact Generalization of Association Schemes

TL;DR

This paper introduces compact association schemes, extending classical finite schemes to a continuous setting, linking algebraic combinatorics with harmonic analysis on compact spaces.

Contribution

It defines compact association schemes that generalize finite schemes and include Voit's compact strong schemes, bridging algebraic combinatorics and harmonic analysis.

Findings

Defines a new class of compact association schemes

Establishes a framework connecting schemes with harmonic analysis

Includes Voit's compact strong schemes within this framework

Abstract

We introduce a notion of compact association schemes, which serves as a compact analogue of classical (finite) association schemes. Our definition is formulated in a way that closely parallels the finite case, naturally admits a Bose--Mesner algebra, and includes the compact strong continuous association schemes introduced by Voit [J. Aust. Math. (2019)] within the framework of hypergroups. This approach provides a new perspective that bridges the theory of association schemes with harmonic analysis on compact homogeneous spaces.