Nonexistence of Certain Skew-symmetric Amorphous Association Schemes

TL;DR

This paper proves that skew-symmetric amorphous association schemes with four or more classes do not exist and shows that all non-symmetric amorphous schemes are commutative, extending the classification of amorphous schemes.

Contribution

It establishes the nonexistence of certain skew-symmetric amorphous schemes and proves that non-symmetric amorphous schemes are necessarily commutative.

Findings

Skew-symmetric amorphous schemes with ≥4 classes do not exist.

Non-symmetric amorphous schemes are commutative.

Classification of amorphous schemes is extended.

Abstract

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.