Four-class Skew-symmetric Association Schemes

TL;DR

This paper classifies and characterizes 4-class skew-symmetric association schemes, determining their intersection matrices, character tables, and feasible parameters, and explores their relation to known schemes and nonexistence results.

Contribution

It provides a complete classification of 4-class skew-symmetric association schemes, including character tables, intersection matrices, and nonexistence proofs for certain schemes.

Findings

Character tables fall into three types.

Determined intersection matrices for these schemes.

Proved no 2-class Johnson scheme admits a 4-class skew-symmetric fission scheme.

Abstract

An association scheme is called skew-symmetric if it has no symmetric adjacency relations other than the diagonal one. In this paper, we study 4-class skew-symmetric association schemes. In J. Ma [On the nonexistence of skew-symmetric amorphous association schemes, submitted for publication], we discovered that their character tables fall into three types. We now determine their intersection matrices. We then determine the character tables and intersection numbers for 4-class skew-symmetric pseudocyclic association schemes, the only known examples of which are cyclotomic schemes. As a result, we answer a question raised by S. Y. Song [Commutative association schemes whose symmetrizations have two classes, J. Algebraic Combin. 5(1) 47-55, 1996]. We characterize and classify 4-class imprimitive skew-symmetric association schemes. We also prove that no 2-class Johnson scheme can admit a 4-class skew-symmetric fission scheme. Based on three types of character tables above, a short list of feasible parameters is generated.