Table algebras of rank 3 and its applications to strongly regular graphs

摘要

A table algebra is called quasi self-dual if there exists a permutation on the set of primitive idempotents under which any Krein parameter is equal to its corresponding structure constants. In this paper we investigate the question of when a table algebra of rank 3 is quasi self-dual. As a direct consequence we find necessary and sufficient conditions for the Bose-Mesner algebra of a given strongly regular graph to be quasi self-dual. In fact, our result generalizes the well-known Delsarte's characterization of a self-duality of the Bose-Mesner algebra of a strongly regular graph given in [P. Delsarte, An algebraic approach to the association schemes of coding theory, Philips Res. Rep. Suppl. 10 (1973) 1-97]. Among our results we determine conditions under which the Krein parameters of an integral table algebra of rank 3 are non-negative rational numbers.