Table algebras with central fusions

The character theory of table algebras is not as good as the character theory of finite groups. We introduce the notion of a table algebra with a central-fusion, in which the character theory has better properties. We study conditions under which a table algebra (A,B) has a central-fusion, and its central-fusion is exactly isomorphic to the wreath product of the central-fusion of a quotient table algebra of (A,B) and another table algebra. As a consequence, we obtain a complete characterization of table algebras with exactly one irreducible character whose degree and multiplicity are not equal. Applications to association schemes are also discussed.