Idempotent systems and character algebras

TL;DR

This paper classifies idempotent systems and symmetric idempotent systems, exploring their structure and relation to character algebras, motivated by association schemes in linear algebra.

Contribution

It provides a classification of idempotent systems and symmetric ones, establishing their connection to character algebras.

Findings

Classification of idempotent systems up to isomorphism

Classification of symmetric idempotent systems

Relationship between symmetric idempotent systems and character algebras

Abstract

We recently introduced the notion of an idempotent system. This linear algebraic object is motivated by the structure of an association scheme. There is a type of idempotent system, said to be symmetric. In the present paper we classify up to isomorphism the idempotent systems and the symmetric idempotent systems. We also describe how symmetric idempotent systems are related to character algebras.