Irreducible characters of wreath products in reality-based algebras and applications to association schemes

Wreath products in reality-based algebras are generalizations of wreath products of table algebras and generalized Camina–Frobenius pairs of C-algebras. In this paper we present characterizations of the wreath product in a reality-based algebra by its irreducible characters and by the size of the zero submatrix of its character table. Applications to finite groups, table algebras, and association schemes are also discussed. In particular, we will show that the wreath product of one-class association schemes is characterized by the zeros in its first eigenmatrix.