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Uniformity in Association schemes and Coherent Configurations : Cometric Q-Antipodal Schemes and Linked Systems

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  • van Dam, E.R.

    (Tilburg University, Center For Economic Research)

  • Martin, W.J.
  • Muzychuk, M.

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.
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Suggested Citation

  • van Dam, E.R. & Martin, W.J. & Muzychuk, M., 2010. "Uniformity in Association schemes and Coherent Configurations : Cometric Q-Antipodal Schemes and Linked Systems," Discussion Paper 2010-10, Tilburg University, Center for Economic Research.
    • Handle: RePEc:tiu:tiucen:3f994561-285b-4be1-88e5-7a1cfd7438dc
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    1. van Dam, E.R., 1997. "Three-Class Association Schemes," Research Memorandum 744, Tilburg University, School of Economics and Management.
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