On structures of quasi-thin table algebras and applications to association schemes

In this paper we develop a structure theory for a class of table algebras, i.e. the so-called quasi-thin table algebras. They are direct generalizations of quasi-thin association schemes. Although this class of table algebras is rather close to finite groups, its structural properties are quite different from those of finite groups. The thin radical and thin residue give rise to two group structures for every table algebra. They play the central role in our discussions. Applications to association schemes are also discussed, and some new results about quasi-thin association schemes are obtained. In particular, we will give necessary and sufficient conditions under which a commutative quasi-thin association scheme has multiplicities 1 or 2. We will need these results in the study of the characterization and classification of double quasi-thin association schemes in [22].