Nonexistence of Certain Skew-symmetric Amorphous Association Schemes

An association scheme is amorphous if it has as many fusion schemes as possible. Symmetric amorphous schemes were classified by A. V. Ivanov [A. V. Ivanov, Amorphous cellular rings II, in Investigations in algebraic theory of combinatorial objects, pages 39–49. VNIISI, Moscow, Institute for System Studies, 1985] and commutative amorphous schemes were classified by T. Ito, A. Munemasa and M. Yamada [T. Ito, A. Munemasa and M. Yamada, Amorphous association schemes over the Galois rings of characteristic 4, European J. Combin., 12(1991), 513–526]. A scheme is called skew-symmetric if the diagonal relation is the only symmetric relation. We prove the nonexistence of skew-symmetric amorphous schemes with at least 4 classes. We also prove that non-symmetric amorphous schemes are commutative.