43,794 research outputs found

    Characterization of Balanced Coherent Configurations

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    Let GG be a group acting on a finite set Ω\Omega. Then GG acts on Ω×Ω\Omega\times \Omega by its entry-wise action and its orbits form the basis relations of a coherent configuration (or shortly scheme). Our concern is to consider what follows from the assumption that the number of orbits of GG on Ωi×Ωj\Omega_i\times \Omega_j is constant whenever Ωi\Omega_i and Ωj\Omega_j are orbits of GG on Ω\Omega. One can conclude from the assumption that the actions of GG on Ωi{\Omega_i}'s have the same permutation character and are not necessarily equivalent. From this viewpoint one may ask how many inequivalent actions of a given group with the same permutation character there exist. In this article we will approach to this question by a purely combinatorial method in terms of schemes and investigate the following topics: (i) balanced schemes and their central primitive idempotents, (ii) characterization of reduced balanced schemes

    A characterization of Q-polynomial association schemes

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    We prove a necessary and sufficient condition for a symmetric association scheme to be a Q-polynomial scheme.Comment: 8 pages, no figur

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

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    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 201
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