Standard Character Condition for C-algebras

TL;DR

This paper introduces the concept of standard character for C-algebras, explores their properties, and characterizes when a commutative table algebra arises from an association scheme.

Contribution

It defines the standard character condition for C-algebras and provides criteria for when such algebras relate to association schemes.

Findings

C-algebras with standard character condition have specific properties

Necessary and sufficient conditions for C-algebras to come from association schemes

Identification of homomorphic images of adjacency algebras

Abstract

It is well known that the adjacency algebra of an association scheme has the standard character. In this paper we first define the concept of standard character for C-algebras and we say that a C-algebra has the {\it standard character condition} if it has the standard character. Then we investigate some properties of C-algebras which have the standard character condition and prove that under some conditions a C-algebra has an adjacency algebra homomorphic image. In particular, we obtain a necessary and sufficient condition for which a commutative table algebra comes from an association scheme.