On Even Generalized Table Algebras

Generalized table algebras were introduced in Arad, Fisman and Muzychuk (Israel J. Math. 114 (1999), 29–60) as an axiomatic closure of some algebraic properties of the Bose-Mesner algebras of association schemes. In this note we show that if all non-trivial degrees of a generalized integral table algebra are even, then the number of real basic elements of the algebra is bounded from below (Theorem 2.2). As a consequence we obtain some interesting facts about association schemes the non-trivial valencies of which are even. For example, we proved that if all non-identical relations of an association scheme have the same valency which is even, then the scheme is symmetric.