Permutation modules over table algebras and applications to association schemes

Inspired by the notion of an action of a finite hypergroup on a finite set we introduce the more general concept of a permutation module over a table algebra. It is easy to see that each permutation module over a table algebra is a direct sum of transitive permutation modules. Among the transitive permutation modules the w-maximal and the maximal permutation modules are the most interesting ones. We give various characterizations of these modules, and we shall see that a standard table algebra admits a maximal transitive permutation module if and only if it arises from a finite association scheme. We also show that the regular module of a standard table algebra is isomorphic to a direct summand of each w-maximal transitive permutation module. As a consequence, one obtains χ(1)⩽mχ for any irreducible character χ of the Bose–Mesner algebra of a finite association scheme and its multiplicity mχ.