Standard Character Condition for Table Algebras

It is well known that the complex adjacency algebra A of an association scheme has a specific module, namely the standard module, that contains the regular module of A as a submodule. The character afforded by the standard module is called 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 standard character condition if it admits the standard character. Among other results we acquire a necessary and sufficient condition for a table algebra to originate from an association scheme. Finally, we prove that given a C-algebra admits the standard character and its all degrees are integers if and only if so its dual.