Functoriality of Bose-Mesner algebras and profinite association schemes

TL;DR

This paper explores the functorial properties of Bose-Mesner algebras in commutative association schemes, extends the theory to profinite schemes, and connects these ideas with combinatorial and number-theoretic structures like nets and sequences.

Contribution

It introduces a functorial perspective on primitive idempotents, develops a Delsarte theory for profinite schemes, and links these concepts with combinatorial designs and sequences.

Findings

Primitive idempotents form a functor from schemes to finite sets.

Profinite association schemes generalize finite schemes to infinite contexts.

Connections with $(t,m,s)$-nets and $(t,s)$-sequences are established.

Abstract

We show that taking the set of primitive idempotents of commutative association schemes is a functor from the category of commutative association schemes with surjective morphisms to the category of finite sets with surjective partial functions. We then consider projective systems of commutative association schemes consisting of surjections (which we call profinite association schemes), for which Bose-Mesner algebra is defined, and describe a Delsarte theory on such schemes. This is another method for generalizing association schemes to those on infinite sets, related with the approach by Barg and Skriganov. Relation with $(t,m,s)$-nets and $(t,s)$-sequences is studied. We reprove some of the results of Martin-Stinson from this viewpoint.