The synchronisation hierarchy via coherent configurations

TL;DR

This paper introduces a combinatorial framework using coherent configurations to analyze the synchronization hierarchy in permutation groups, leading to new insights and computational tools for group properties.

Contribution

It establishes a novel connection between the synchronization hierarchy and coherent configurations, enabling more efficient analysis and classification of permutation groups.

Findings

Every spreading permutation group of degree ≤ 8191 is a QI-group.

The framework unifies properties in the synchronization hierarchy.

Enhanced computational methods for analyzing permutation groups.

Abstract

We describe the spreading property for finite transitive permutation groups in terms of properties of their associated coherent configurations, in much the same way that separating and synchronising groups can be described via properties of their orbital graphs. We also show how the other properties in the synchronisation hierarchy naturally fit inside this framework. This combinatorial description allows for more efficient computational tools, and we deduce that every spreading permutation group of degree at most $8191$ is a $\mathbb{Q}$I-group. We also consider design-orthogonality more generally for noncommutative homogeneous coherent configurations.