Mathematics – Combinatorics
Scientific paper
2010-01-27
Mathematics
Combinatorics
40 pages, 1 figure, 1 table
Scientific paper
Inspired by some intriguing examples, we study uniform association schemes and uniform coherent configurations. Perhaps the most important subclass of these objects is the class of cometric Q-antipodal association schemes. The concept of a cometric association scheme is well-known; however, until recently it has not been studied well outside the area of distance-regular graphs. Uniformity is a concept introduced by Higman, but this likewise has not been well-studied. 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 obtain additional information. 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 examine the consequences of this equivalence. 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 feasible parameter sets for four-class schemes.
Martin William J.
Muzychuk Mikhail E.
van Dam Edwin R.
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