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

Understand

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.