A construction of pairs of non-commutative rank 8 association schemes from non-symmetric rank 3 association schemes

Abstract

We construct a pair of non-commutative rank 8 association schemes from a rank 3 non-symmetric association scheme. For the pair, two association schemes have the same character table but different Frobenius–Schur indicators. This situation is similar to that for the dihedral and quaternion groups of order 8. We also determine the structures of adjacency algebras of the rank 8 schemes over the rational number field.