The non-existence of certain skew-symmetric amorphous association schemes

An association scheme is amorphous if it has as many fusion schemes as possible. At the algebraic level, symmetric amorphous schemes were classified by Ivanov (1985) [12] and commutative amorphous schemes were classified by Ito et al. (1991) [11]. A scheme is called skew-symmetric if the diagonal relation is the only symmetric relation. We show the non-existence of skew-symmetric amorphous schemes with at least four classes. We also prove that non-symmetric amorphous schemes are commutative.