A characterization of exceptional pseudocyclic association schemes by multidimensional intersection numbers

TL;DR

This paper characterizes exceptional pseudocyclic association schemes derived from specific permutation groups using multidimensional intersection numbers, providing a new classification approach for these mathematical structures.

Contribution

It introduces a novel characterization of exceptional pseudocyclic schemes through multidimensional intersection numbers, extending understanding of their structure beyond known cases.

Findings

Most exceptional pseudocyclic schemes are uniquely determined by their 3-dimensional intersection numbers.

Three specific cases are exceptions to this characterization.

The work links group actions to the combinatorial properties of association schemes.

Abstract

Recent classification of $\frac{3}{2}$-transitive permutation groups leaves us with three infinite families of groups which are neither $2$-transitive, nor Frobenius, nor one-dimensional affine. The groups of the first two families correspond to special actions of ${\mathrm{PSL}}(2,q)$ and ${\mathrm{P\Gamma L}}(2,q),$ whereas those of the third family are the affine solvable subgroups of ${\mathrm{AGL}}(2,q)$ found by D. Passman in 1967. The association schemes of the groups in each of these families are known to be pseudocyclic. It is proved that apart from three particular cases, each of these exceptional pseudocyclic schemes is characterized up to isomorphism by the tensor of its $3$-dimensional intersection numbers.