Uniformity in association schemes and coherent configurations
| Autor: | William J. Martin, Edwin van Dam, Mikhail Muzychuk |
|---|---|
| Přispěvatelé: | Research Group: Operations Research, Econometrics and Operations Research |
| Jazyk: | angličtina |
| Rok vydání: | 2013 |
| Předmět: |
Discrete mathematics
Strongly regular graph Pure mathematics Antipodal point Fiber size Context (language use) Characterization (mathematics) Graph Theoretical Computer Science Association scheme Computational Theory and Mathematics If and only if Scheme (mathematics) FOS: Mathematics Mathematics - Combinatorics Discrete Mathematics and Combinatorics Combinatorics (math.CO) Indecomposable module 05E30 (Primary) 05B25 05C50 51E12 (Secondary) Mathematics |
| Zdroj: | Journal of Combinatorial Theory, Series A, Structures designs and application combinatorics, 120(7), 1401-1439. Academic Press Inc. |
| ISSN: | 0097-3165 |
| DOI: | 10.1016/j.jcta.2013.04.004 |
| Popis: | 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. Comment: 42 pages, 1 figure, 1 table. Published version, minor revisions, April 2013 |
| Databáze: | OpenAIRE |
| Externí odkaz: |
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