Table algebras with exactly one irreducible character whose degree and multiplicity are not equal

Commutative standard table algebras with exactly one multiplicity not equal to 1 are characterized by the wreath product of some special table algebras in [1]. A natural and much more general question is the characterization of standard table algebras (not necessarily commutative) with exactly one irreducible character whose degree and multiplicity are not equal and the degree is 1. We will give a characterization of such table algebras, including the main result of [1] as a special case. Applications to association schemes are also discussed.