TY - JOUR

T1 - Uniformity in association schemes and coherent configurations

T2 - Cometric Q-antipodal schemes and linked systems

AU - Van Dam, Edwin R.

AU - Martin, William J.

AU - Muzychuk, Mikhail

N1 - Funding Information:
E-mail addresses: Edwin.vanDam@uvt.nl (E.R. van Dam), martin@wpi.edu (W.J. Martin), muzy@netanya.ac.il (M. Muzychuk). 1 The author was supported in part by NSA grant number H98230-07-1-0025.

PY - 2013/9/1

Y1 - 2013/9/1

N2 - 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.

AB - 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.

KW - Coherent configuration

KW - Cometric association scheme

KW - Imprimitivity

KW - Linked system

KW - Q-antipodal association scheme

KW - Strongly regular graph decomposition

KW - Uniform association scheme

UR - http://www.scopus.com/inward/record.url?scp=84876710747&partnerID=8YFLogxK

U2 - 10.1016/j.jcta.2013.04.004

DO - 10.1016/j.jcta.2013.04.004

M3 - Article

AN - SCOPUS:84876710747

VL - 120

SP - 1401

EP - 1439

JO - Journal of Combinatorial Theory - Series A

JF - Journal of Combinatorial Theory - Series A

SN - 0097-3165

IS - 7

ER -