Uniformity in association schemes and coherent configurations: cometric Q-antipodal schemes and linked systems

TL;DR

This paper explores uniform association schemes and coherent configurations, focusing on cometric Q-antipodal schemes, providing characterizations, parameter relations, and connections to designs, graphs, and geometries.

Contribution

It offers new characterizations of uniform schemes, links cometric Q-antipodal schemes to designs and graphs, and classifies feasible parameters for certain schemes.

Findings

Uniform schemes are equivalent to dismantlable schemes.

Cometric Q-antipodal schemes are characterized by uniformity.

Examples include linked systems of symmetric designs and strongly regular graphs.

Abstract

Inspired by some intriguing examples, we study uniform association schemes and uniform coherent configurations, including cometric Q-antipodal association schemes. After a review of imprimitivity, we show that an imprimitive association scheme is uniform if and only if it is dismantlable, and we cast these schemes in the broader context of certain --- uniform --- coherent configurations. We also give a third characterization of uniform schemes in terms of the Krein parameters, and derive information on the primitive idempotents of such a scheme. In the second half of the paper, we apply these results to cometric association schemes. We show that each such scheme is uniform if and only if it is Q-antipodal, and derive results on the parameters of the subschemes and dismantled schemes of cometric Q-antipodal schemes. We revisit the correspondence between uniform indecomposable three-class schemes and linked systems of symmetric designs, and show that these are cometric Q-antipodal. We obtain a characterization of cometric Q-antipodal four-class schemes in terms of only a few parameters, and show that any strongly regular graph with a ("non-exceptional") strongly regular decomposition gives rise to such a scheme. Hemisystems in generalized quadrangles provide interesting examples of such decompositions. We finish with a short discussion of five-class schemes as well as a list of all feasible parameter sets for cometric Q-antipodal four-class schemes with at most six fibres and fibre size at most 2000, and describe the known examples. Most of these examples are related to groups, codes, and geometries.