Functors from Association Schemes

TL;DR

This paper develops a specialized subcategory of association schemes with properties enabling a group-like isomorphism theorem and functorial constructions, providing a conceptual framework for character product results.

Contribution

It introduces a new subcategory of association schemes with a group-like structure and functorial properties, enhancing the understanding of scheme-to-group and scheme-to-algebra mappings.

Findings

Established a first isomorphism theorem for the subcategory

Demonstrated functoriality of schemes to groups and algebras

Provided a conceptual account for Hanaki's product of characters

Abstract

We construct a wide subcategory of the category of finite association schemes with a collection of desirable properties. Our subcategory has a first isomorphism theorem analogous to that of groups. Also, standard constructions taking schemes to groups (thin radicals and thin quotients) or algebras (adjacency algebras) become functorial when restricted to our category. We use our category to give a more conceptual account for a result of Hanaki concerning products of characters of association schemes; i.e. we show that the virtual representations of an association scheme form a module over the representation ring of the thin quotient of the association scheme.