Generalized table algebras

A table algebra was defined in [1] in order to consider in a uniform way th e common properties of conjugacy classes and irreducible characters. Non-co mmutative table algebras were introduced in [5]. They generalize properties of such well-known objects as coherent and Hecke algebras, Here we extend the main definition of a non-commutative table algebra by letting the groun d field be an integral domain. We call these algebras generalized table alg ebras (GT-algebras, in brief). It is worth mentioning that this class of al gebras includes generic Hecke-Iwahori algebras of finite Coxeter groups. We develop the general theory for this type of algebras which includes their representation theory and theory of closed subsets. We also study the prope rties of primitive integral table algebras.