Schurity and separability of quasiregular coherent configurations

TL;DR

This paper investigates when quasiregular coherent configurations are schurian and separable, establishing duality results and generalizing conditions involving distributive lattices of normal subgroups.

Contribution

It establishes a duality for quasiregular coherent configurations and generalizes conditions for schurity and separability based on group lattice properties.

Findings

Duality between schurity and separability in quasiregular coherent configurations.

Generalization of conditions for schurity and separability involving distributive lattices.

Extension of Hanna Neumann's results to broader classes of configurations.

Abstract

A permutation group is said to be quasiregular if every its transitive constituent is regular, and a quasiregular coherent configuration can be thought as a combinatorial analog of such a group: the transitive constituents are replaced by the homogeneous components. In this paper, we are interested in the question when the configuration is schurian, i.e., formed by the orbitals of a permutation group, or/and separable, i.e., uniquely determined by the intersection numbers. In these terms, an old result of Hanna Neumann is, in a sense, dual to the statement that the quasiregular coherent configurations with cyclic homogeneous components are schurian. In the present paper, we (a) establish the duality in a precise form and (b) generalize the latter result by proving that a quasiregular coherent configuration is schurian and separable if the groups associated with homogeneous components have distributive lattices of normal subgroups.